Gas blower won't stay started

What year and model?

Check air filter is clean , jump out pressure switch see if it lights

The damage is already done before you put the fuel filter in. (dirt in Carb)
Remove your carb and dissesemble it and clean everything out and then put back on the machine with FRESH gas and you will be fine...run like a top !

clearly sounds like fuel starvation, I would be looking at the throttle body, they need servicing every couple of years, they fill up with like grease and block tick over hole.....

Hold the power button in for a few seconds it will shut down. Then unplug system and push the power button down with it unplugged then replug it in and power up again. Then do a proper shut down again to see if the problem fixed. I hope this assisted you. John

If your furnace lights and the gas stays on for 8 to 10 seconds, then shuts right back off, then you need to clean your flame sensor with light sand paper or steel wool. You might need a new flame sensor, but most of the time they can be cleaned an will work well after cleaning.

There is a flame sensor failure.

Here is the ignition sequence for a furnace. This will help you understand what is happening/not happening.

Thermostat calls for heat.

draft inducer motor (DIM) starts drawing draft through heat exchanger.

Pressure switch detects proper operation of DIM, and sends signal to board to begin ignition.
(it detects negative pressure)

Ignitor starts ignition attempt and gas valve sends gas to burners.

Upon ignition, flame sensor detects flame and sends signal to board stopping ignition sequence and continuing gas flow for heat mode operation until thermostat is satisfied.

During this sequence the blower motor is brought on, and by using a series of temperature limit switches, the system assures that the unit is operating properly.
(If the blower is not operating, or the airflow is inadequate, the heat exchanger will overheat and trip one or more temp limit switches.)

There is also a "rollout" switch that detects improper flame pattern "rolling out" into the controls compartment.

Now, to apply that to your situation.

If the flame sensor does not sense flame, it shuts down the furnace to prevent a dangerouse gas build up.

With most modern glow coil ignition systems you have one of two failures.

1) the flame sensor is bad and needs to be replaced.

2) the control board is not proccesing the signal from the flame sensor and shuts the system down. In that case the board would need to be replaced.

There is a possibility that a temperature limit or roll out switch is breaking the curcuit, but that would be quite unusual to happen consistantly at three seconds.

Unless you have the ability and tools to test the snesor and board, the next step would be to replace the sensor first, and then the board if that does not fix it.

In order to accurately determine and diagnose the problem, a more detailed description of the problem needs to be communicated as opposed to normal operation of the system so a repair person can determine that there is in fact a problem. There's a big difference between solving a problem via text as opposed to seeing what is wrong in the field. They rarely are the same.

Goodman condenser sequence of operation on a matched system. Call for cool, condenser starts, 15-30 seconds later blower motor starts, cool satisfied, condenser shuts down, 15-45 seconds later blower motor shuts down. House heats ups and a call for cool via the thermostat starts another cycle.

Based on normal operation it is impossible to determine whats wrong with the information given. There are perhaps 5 voltage measurements in the system that you could be referring to.