Why does my wall unit never stop blowing air when set to AC?

It sounds like your PTAC (packaged terminal air conditioner) unit is malfunctioning. When the thermostat is set to heating mode, the PTAC should be blowing warm air, not cold air. There are a few possible causes for this issue:

1. The PTAC may be low on refrigerant. If the unit is low on refrigerant, it may not be able to properly heat the air.
2. The PTAC's reversing valve may be stuck in the cooling position. The reversing valve is responsible for switching the flow of refrigerant between heating and cooling mode. If it is stuck in the cooling position, the PTAC will blow cold air even when the thermostat is set to heating mode.
3. The PTAC's heating element may be malfunctioning. If the heating element is not functioning properly, the PTAC will not be able to heat the air.

I recommend contacting a trained HVAC technician to diagnose and fix the issue with your PTAC. They will be able to determine the cause of the problem and repair it properly.

Is this an electric unit or does it run on fuel?

When you mansion water in the pan, do you mean the indoor coil pan or the outdoor catch "pan"? I've seen some models have a drain plug that's on the bottom of the outdoor panel or underneath. And that plug should remain installed. It uses that water to cool the copper freon tube that curves along the bottom of the outdoor section. With plug removed it can overheat. Basically it does not drain onto the ground outside, it retains its water. And the air entering side of outdoor coil is clean? May sound like repetition(forgive me). The side you see clearly is the air leaving side. The dual shaft motor has a run capacitor that may be weakening. And the outdoor air that passes thru the outdoor coil may be recirculating out then directly back in to the unit causing it to overheat. Anything blocking airflow outside? 10 mins of runtime then shouting down tells me its overheating. It may even be the electrical circuit. Loose connections can cause high amperage, low voltage issue and cause the ac load to over amp. Over amp=over heat. Hope this helps!

Most air conditioners now are reverse cycle, which means that they can heat as well as cool. Cooling only units were popular many years ago due to the price difference for a reverse cycle. Today, the cost of manufacturing has dropped and it has resulted in cooling only units being virtually finished. I would suggest you get a local air con sales company to come to your house to measure up properly and determine what would be best. There are a lot of factors in installing air conditioning. IE/ The size of the house, the area it will be required to heat / cool, the amount of windows, types of walls, roof etc (Heat loading on the house) Roof and wall insulation etc etc.

From what you describe it sounds like it might be a relay problem.
It needs to be diagnosed properly first.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/dan_77d17b8918f54455

Has any one checked the exhaust vent,other than a bad thermostat the number one cause for the thermal fuse to open is a clogged or restricted exhaust vent flow, the entire vent system needs to be checked ( not just the hose from the unit to the wall) for proper air flow. Turn the unit on TIMED ,AIR DRY (NO HEAT) and go outside to the exhaust vent and check the air flow,if there is any question as to weather the air flow is sufishent get someone out to clean the exhaust vent ,it always a good idea to clean out every few years regardless to help prevent build up of lint witch could cause a fire......Hope this Helps Good Luck

Did you clean the exhaust from the wall behind your dryer to the outside. This is more important than the dryer exhaust system itself. If not get behind dryer and remove exhaust to the wall and using a leaf blower,(electric) put it inside house vent then push dryer back to hold in place go outside and watch to see what all blows out. Good luck.

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.

Setting temps on a Wall unit or a window unit - depends on where the Unit is in the room ( area), the stat in the unit is reacting to the the return air ( at the Unit) . If the Unit is not large /small enough then you will have a problem - 1. To Cold near the Unit and to warm the further away you are from it.
2. If the Unit is facing a wall to close, the unit can short cycle ( entire room will be warm). Dedending on the room and heat load - check and see if the Unit is the proper size. " to much of a good thing " is bad - not enough maybe to ( not cooling the area properly). #3 Control is defected ( age of Unit ? )
was not given.

A "Heat-Pump" is also known as a reverse cycle air conditioner, this means on heat the outdoor unit acts like the indoor section and vice versa. So in all probability you need to have it leak checked and recharged.