Furnace runs on heat, air conditioner does not

The fan motor is thermally protected the motor seems to be cutting out on the overload replace the fan motor

Need furnace model number. It sounds like you're looking for a standing pilot. Your furnace might not have one. Very likely to have a spark ignition type system or hot surface igniter.

MOST UNITS NOT HEAT PUMPS  WIRE UP WITH 2 WIRES TO CONDENSER RED WIRE AND WHITE . WHITE GOES TO COMMON C ON FURNACE AND RED TO Y TERMINAL OF FURNACE Y ALSO GOES TO Y ON TSTAT. BOTH YELLOW WIRES MUST MEET AT Y OF FURNACE.  WHITE FROM STAT TO W   GREEN FROM TSTAT TO G AT FURNACE IF YOU HAVE C ON TSTAT HOOK UP BROWN WIRE OR BLUE FROM COMMON TERMINAL WHERE YOU HOOKED UP CONDENSER WHITE WIRE AT FURNACE TO TSTAT , MOST THERMOSTATS DONT NEED C RUN FROM FURNACE. THE TWO WIRES GOING TO CONDENSER IS -  AND + TO ACTAVATE CONTACTOR IN OUTSIDE UNIT

If it does not have a furnace or heating elements for heating, then it is a heat pump, which is rare for a central AC system, but makes sense of the fact that it blows warm air when you reset the pressure switch, which may indicates freon pressure charge problem if your AC system has a reverse valve and uses freon to for heating (heat Pump).

Do you get heating when outside unit runs? if you do, check freon pressure levels and top up to required pressures. The button you depress is a pressure reset switch ... that is if your AC system is a heat pump ... which, as pointed above, it is not common for central ACs.

nope, you're sol - have to replace evaporator coil assembly - sorry for the bad news

Tell him to make tight connections.

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Check to see if the green thermostat wire is conneceted to the "G" terminals on the thermostat and on the furnace.

You,ll need to open the furnace and observe the led lights on the comtrol board the is a chart attaced to cover. If it is in lock out mode the fan runs. To reset turn off at tstat.the turn the power on at the furnace, set tsat up, observe furnace. First you should hear a small fan then the glow of the ignitor then the opening of the gas valve and flames 10-30 seconds then the big fan.

It sounds like the unit is tripping off from high pressure. Check you filter and indoor coil. You may have low air flow inside that is causing the refrigerant pressure to be to high.

check the filter also sounds like a low freon charge.