Fan was running on low speed, sped up to high by it self and now cannot turn of or change speed with remote. Changed batteries and have turned fan on and off at fusebox - with same result.

Change the batteries in the remote for brand new Duracell.

Both radiator cooling fans are running all the time ? Key off ? The cooling fan opperation
HOW they work
Cooling Fan Control
The engine cooling fan system consists of 2 cooling fans and 3 relays. The relays are powered by the battery positive voltage circuit and controlled by a switched ground from the powertrain control module (PCM).
During low speed operation, the PCM supplies the ground path for the low speed cooling fan relay through the low speed cooling fan relay control circuit. This energizes the cooling fan low relay coil, closes the relay contacts, and supplies battery positive voltage from the COOL FAN LO fuse through the cooling fan motor supply voltage circuit to the cooling fan. During high speed operation the PCM supplies the ground path for the cooling fan low relay through the low speed cooling fan relay control circuit. After a 3-second delay, the PCM supplies a ground path for the cooling fan high relay and the s/p cooling fan relay through the high speed fan relay control circuit.
The PCM commands the fan on under the following conditions:
• Engine coolant temperature exceeds approximately 98°C (208°F) Low Fan Speed
• Engine coolant temperature exceeds approximately 102°C (216°F) High Fan Speed
• A/C refrigerant pressure exceeds 361 kPa (52 psi) Low Fan Speed
• A/C refrigerant pressure exceeds 2100 kPa (300 psi) High Fan Speed
• When the engine coolant temperature exceeds 112°C (234°F) at key off, the fan high speed will run for up to 300 seconds. If within that time frame 102°C (216°F) is reached then fan speed will change from high to low speed. If within that time frame 99°C (210°F) is reached then fan speed will change from low to off.
The PCM commands the fan off under the following conditions:
• A/C is requested and engine speed exceeds 6,240 RPM
• Engine coolant temperature exceeds approximately 99°C (210°F) turns the cooling fans from low to off.

sounds like one of the relays are stuck closed .

Cooling Fan Always On
Step
Action
Yes
No
Schematic Reference: Engine Cooling Schematics , HVAC Schematics in HVAC Systems - Manual
Connector End View Reference: Cooling System Connector End Views
DEFINITION: One or both engine cooling fan motors run continuously in high or low speed.
1
Did you perform the Diagnostic System Check - Vehicle?
Go to Step 2
Go to Diagnostic System Check - Vehicle in Vehicle DTC Information
2
Install a scan tool.
Turn ON the ignition, with the engine OFF.
With a scan tool, command the Fans Low Speed ON and OFF.
Are one or both cooling fans ON?
Go to Step 3
Go to Testing for Intermittent Conditions and Poor Connections in Wiring Systems
3
Are both cooling fans running continuously?
Go to Step 4
Go to Step 6
4
Are both cooling fans running continuously in high speed?
Go to Step 5
Go to Step 7
5
Test the low reference circuit of the A/C refrigerant pressure sensor for an open. Refer to Circuit Testing and Wiring Repairs in Wiring Systems.
Did you find and correct the condition?
Go to Step 18
Go to Step 14
6
Important: The S/P relay is located under the front bumper on the drivers side of the vehicle.
Remove the cool fan S/P relay.
Did the right cooling fan turn OFF?
Go to Step 10
Go to Step 8
7
Remove the cool fan LOW relay.
Did the cooling fans turn OFF?

The engine cooling fan system is controlled by the body control module (BCM) and the powertrain control module (PCM) or engine control module (ECM). The BCM performs the calculations as to how long, when and what speed the cooling fans should turn on. The BCM then sends a class 2 message to the PCM/ECM to engage the cooling fan relays. If there is a malfunction with the BCM, the PCM/ECM will control the engine cooling fans independently. The engine cooling system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the PCM/ECM to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood accessory wiring junction block. The ground path is provided at G100.
During low speed operation, the PCM/ECM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the cooling fan 1 relay coil, closes the relay contacts, and supplies battery positive voltage from the cool fan 1 fuse through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the cooling fan 2 relay and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation the PCM/ECM supplies the ground path for the cooling fan 1 relay through the low speed cooling fan relay control circuit. After a 3 second delay, the PCM/ECM supplies a ground path for the cooling fan 3 or S/P relay and the cooling fan 2 relay through the high speed cooling fan relay control circuit. This energizes the cooling fan 2 relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the cooling fan 2 relay coil is energized closing the relay contacts and provides battery positive voltage from the cool fan 2 fuse on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.
The BCM and PCM/ECM command Low Speed Fans on under the following conditions:
?€¢
Engine coolant temperature exceeds approximately 106?°C (223?°F).

?€¢
A/C refrigerant pressure exceeds 1 310 kPa (190 psi).

?€¢
After the vehicle is shut off if the engine coolant temperature at key-off is greater than 140?°C (284?°F) and system voltage is more than 12 volts. The fans will stay on for approximately 3 minutes.

The BCM and PCM/ECM command High Speed Fans on under the following conditions:
?€¢
Engine coolant temperature reaches 110?°C (230?°F).

?€¢
A/C refrigerant pressure exceeds 1 655 kPa (240 psi).

?€¢
When certain DTCs set.

The three relays are located in the under hood fuse / relay box ! You need to have it checked for DTC'S -diagnostic trouble codes ! An you need a GM Tech 2 scan tool !
This diagnostic procedure supports the following DTCs:
?€¢
DTC P0480 Cooling Fan 1 Relay Control Circuit

?€¢
DTC P0481 Cooling Fan 2 and 3 or S/P Relay Control Circuit

?€¢
DTC P0691 Cooling Fan 1 Relay Control Circuit Low Voltage

?€¢
DTC P0692 Cooling Fan 1 Relay Control Circuit High Voltage

?€¢
DTC P0693 Cooling Fan 2 and 3 or S/P Relay Control Circuit Low Voltage

?€¢
DTC P0694 Cooling Fan 2 and 3 or S/P Relay Control Circuit High Voltage

1. Install a scan tool.
2. Turn ON the ignition, with the engine OFF.
3. With a scan tool, command the Fans Low Speed ON and OFF.

Do the low speed engine cooling fans turn ON and OFF with each command?
Go to Step 3
Go to Step 4

3

Important:: A 3 second delay occurs before the powertrain control module (PCM)/engine control module (ECM) changes the cooling fan speed.
With a scan tool, command the Fans High Speed ON and OFF.
Do the high speed engine cooling fans turn ON and OFF with each command?

That is most likely the unit overheating and trying to cool itself off. Check any filters that are on it to be sure they are clean.

If you have both heat and ac running that means that the Yellow , AC and the White heat circuits are both being energized with the Red 24 volt hot leg that enters the tstat on the R terminal, note the Auto fan switch between Yellow and Green if Normally closed that is the Auto setting and high speed fan comes on for the Cool mode via the fan relay,
while the Heat speed is low speed and controlled not by the fan relay but the temperature or time delay, you need to determine what speed you have running now, low or high? if you flip the fan switch to ON and it speeds up then it was on low, if it makes no difference you have high sped running already.

Check your capacitor.

It sounds like fan motor is okay. There are two motor windings -- one low speed, one medium speed. Low winding gives you low speed, medium winding gives you medium speed, both windings give you high speed -- so I would say if you get high speed, both windings are good.

The remote (the hand held transmitter) controls the speed by raising and lowering the frequency of the current which is 60 cycle coming to the fan ceiling box. If the fan speed -selector switch (usually a pull switch) is set to high, when the remote (transmitter) signals the receiver to change speeds, the frequency of the current is changed by the receiver (the receiver is usually in the fan housing).

Sounds like you'll need to replace the remote system -- these come in sets, both transmitter and receiver, and cost about $30. If you're not experienced with wiring fans, you should get an electrician to do the job.

Under the dome (where the electrical connections are) is the receiver unit for the fan. It could be that it is no longer working. Also the remote may have dip switches to change the frequency channel (sometimes located behind the batteries) check that these have not been changed or check they correspond to the receiver unit on the fan.