Gas and Oil Home Heating Furnaces - Maintenance tips needed

Hi folks, I read through quite a few posts & didn't see anything similar, so here goes:
I have a 3 ton Carrier heat pump for a 2500 sf house on a slab. The unit is about 5 1/2 years old and is located at one end of the house. Last year I had some problems with the low voltage fuse blowing, but after "jiggling" the wires the problem went away. The unit has never blown very strongly and we have a temperature gradient from one end of the house to the other (+/-60'). While woking on the fuse problem, I noticed that the motor is a 3 speed unit and was wired for low speed. When it started to get hot this year, I switched the wires to the high speed setting and it helped with the temperature variation. Lately though, it seemed like the unit is running all the time. I checked the temp of the air on the outflow side and it was only 10 deg cooler than the room air. I took of one of the filters (changed monthly) and looked up into the coils. Egad, they were literally covered with moist dust. The stuff wouldn't vacuum off easily so I used a nylon brush to gently work it loose. They aren't sparkling clean, but are much better. Now when I check the temp I get only 6 degrees of difference???
Is there anything I can spray on the coils to help remove the remaining crud. The coils are in a tent shaped frame, should I wash them from the other side to push the dirt out?

Any tips and suggestions would be appreciated.

Thanks in advance,

Dave in Columbus, MS

One additonal point, the thermistat is located n the wall of the cupboard that houses the inside unit.

A coil cleaning will almost certainly help. There are plenty of sprays that you can get that will do the job. Some require extensive rinsing, others are "self-rinsing". I like rinsing with a pump sprayer or water hose. There's more mess to clean up, but I like the end result better.

Feel the larger of the two copper refrigerent lines. If it's not fairly cool or cold to the touch, you need to have your freon levels checked. I strongly suspect that you are probably undercharged. Make sure that the tech finds the leak. A quick test you can do yourself is to put soapy water on the service valves on the condensing unit outside. A fair amount of the time, the valve cores on those service ports are the problem. They are very easy to replace with the right tool.

And yes, increasing the airflow will result in a smaller temperature differential. That's why you need to clean the coil before having the freon checked.

if your living space is 2500 square feet you are hugely undersized. If that is inclusive of garage space you might be borderline. I install 5 ton units with that footage.