The air comes on and blows, but the fan outside doesn't spin....tho the air feels a bit cool, the apt never cools off

I have a Fujitsu Split unit inverter with a second evaporator in the attic. I am not impressed with them. I have found that disconnecting the power at the mains for a couple of hours, either with the breaker switches off, or the outside isolator effects a good power reset. If you listen at the outside condenser unit, you will hear clicks and rattles as it goes through a test procedure. Don't switch on for 15 minutes.

Hottest weeks of the year and the vent on the inner high level unit fails to open! :>(

Check to see if air is blowing out the back side of condersor,outside ,maybe fan not blowing on outside, or freon low.

check for loose wiring to Tstat, or the new stat has failed.
if the stat has a fan on switch and when on the fan works then replace the stat

The buzzing sounds like the fan trying to start. Try a new fan capacitor, or get a new fan

This is usually a bad capacitor or bad fan motor. The capacitor is located inside the electrical panel on the outside unit. You can check these by using a meter that measures Mfds (Microfarads), most of the time you can visually see the capacitor swollen or blown. The fan motor can be checked by a meter that measures ohms, really high ohms are usually bad. If you can't spin the fan blade easily, then it's locked up. What usually happens, the capacitor or motor windings go bad, doesn't spin the fan blade, overheats the compressor so it kicks out on thermal protection till it cools back down, and sometimes the motor will overheat and go into thermal protection also. Just make sure you are getting the 220v to the outside unit too, but shut all power off before doing any trouble-shooting. Hope this helps, feel free to ask any questions!

your auxillary fan is not working. this fan is responsible for cooling the freon that is being circulated in your cooling system... check if there are any loose wires in front of the engine bay.

the auxillary fan is located right beside your radiator fan.

When you first turn it on the fan runs and after a few minutes the outside unit comes on, once the outside unit comes on it take time for it to cool the coils which cool the house, And a/c does not really cool the air,It removes the heat that is in the air.

change the run capacitor on the condensing fan motor

sound like high pressure switch going out,,, if you replace it ,, you have to replace the gas,, but will cause it to cut off when high pressure hits,, and work again when cooled off

Your ac thermostat is defective and not switching off the compressor resulting in excess cooling and need replacement. Also ensure that all doors and windows are closed. Could be some window is opened behind the curtain without your knowledge. That could be yet another reason for your unit continuously running.
Finally check your relay unit and this should solve your problem!