My air handler runs but does not cut out, once it reaches the temperature set in the thermostat the external unite kicks off but the fan in the air handler does not stop running.Looking for a relay or switch that's hanging up but not sure. Please help.

Thermostats have multiple issues when you start checking them against thermometers.

1) the built-in thermometer may not match the set temperature because one or both need calibration. Compare the built-in to an external thermometer and see if there is any adjustment to make them match. Record the setting on the thermostat and the temperature when it starts and stops -- there should be a 1-3 degree range which may be adjustable. The set point should be in the middle of the on-off temperature spread.
2) the thermostat has a heat anticipator setting which may be set poorly -- turning off the heat before it reaches the set point - the fan normally runs for 1-2 minutes after the heat source is off. The anticipator is used to stop heating before reaching the set point as there are still a lot of BTUs coming out during the fan run-down -- usually 1-2 minutes.

check the operators manual but you will probably find that the fan will keep running so that the air in the room is circulated past the thermostat so that the temperature can be monitored and the compressor switched on when the temp reaches the upper limit

If you have a digital thermostAt see if it has a reset option.
Also loosen from wall and check for a loose connection
Check your air handler for the same( your small thermo wireset)
If there is no reset option and no loose connections
Replace your thermostat

There seems to be a problem with the wiring of the thermostat. Apparently the thermostat is calling for cool (if wired properly), but is not being satisfied. The thermostat is just a switch. When there's a call for cool, the thermostat sends 24-28 volts to the contactor (inside the condensing unit to start the compressor and outdoor fan). The indoor fan is energized through the fan relay (inside the air handler). When the temperature in the conditioned space reaches the set temperature,then the thermostat cuts the power to the contactor and it stops the compressor and outdoor fan. Check that the compressor is actually running, if not, check for 24-28 volts on the wires feeding the contactor, if there is, and the contactor isn't closing then the contactor is defective. If the contactor closes and the compressor doesn't come on, check the capacitor. If the capacitor is good then check your compressor motor. I suspect you have an improperly wired thermostat

your thermostat could be wired in to a energy recovery venilator which will bring the unit on for a certain amount of time every hour some thermostats have recirculate mode which will do the same thing to help filter the air

thermostat most likely cause.

The thermostat is the problem,this is what controls the off,and on cycle for the compressor.

I have a honeywell TH8320U thermostat and American standard Air handler with heating unit put in. THE thermostat works fine with the ac and follows the programs I have set. However the heat is a different story. The thermo clicks on to follow program but the unit does not run. (auto settings) If I place the thermo in heat and turn it to on I get heat?

If you have a standard thermostat the fan switch is for the air handler only. It completely over rides the thermostat for most settings. Now lets see this is what I'd expect:
fan switch on Indoor fan runs all time
Fan switch auto Indoor fan only runs when thermostat is set to cool or heat and the thermostat is turned up above room temp for heat or below room temp for cooling. In both of these modes the furnace blower comes on and stays on til the room temperature is satisfied. The outdoor unit only comes on for AC the out door unit will not come on for heat or the "on" setting on the thermostat. See exception 1 & 2 below...

Exception 1- (in a heat pump the out door unit runs in both heat and cool mode unless it has emergency heat turned on then it will not come on outside in the heat mode, and depending on the type of heat pump and furnace or air handler it is). A heat pump thermostat usually has "backup heat" and an "emergency heat" setting switch on the thermostat.
Exception 2 - On some thermostats they are made to cycle on the heat if the temperature reaches a preset temp (around 50'f usually but definitely by 40'f). This mode prevents freezing when if instance the homeowner is away or while the home is under construction. In this mode the fan inside can run but the heat usually comes on with it.
Exception3 - On some electric furnaces the heating strips may be individually controlled and thus not all come on at one time. This may give the feel of no heat especially if the room is cold.

Now there may be other exceptions but Ive tried to give you a heads up on some of the most common ones.
If you need more help please post more specifics about your unit/system as to for sure what switches are set to what setting and what is happening with each. My first though was that maybe your system is OK and you needed to know if the thermostat was working properly.

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It sounds like a bad or badly miswired T-stat