I completely replaced my entire ac system on my last Silverado 2500hd. Was in plumbing/HVAC.
First your thermostat will be for engine not ac.
Also there is low and high cutoff switches that make the compressor (on which on shut off switch is. I believe low on most trucks). If refrigerant level is too low OR high your compressor will not turn on. First check the pressure via shrader valve on refrigerant line near passenger side back (facing truck front). The high cut off or in some cases maybe low would be near the headlight sometimes frustratingly close to the radiator or otherwise harder to get too. Bumper disassembled is much easier and pulling headlights.
If you have pressure not high or low (I would tell you but it's truck specific just Google your correct refrigerant pressure on line.)
Also don't just buy those kits to "fix your ac" that makes so many compressors just shut off over pressured when it could be so much more.
Replace BOTH cut offs if pressure is ok. Check truck getting it moving helps airflow which is how ac works so it will be easier to tell if any progress is made moving.
(For those interested in how it operates otherwise next paragraph)
If, both sensors tell computer pressure is in range, compressor should kick on and fan. This is also via accessory drive belt meaning listen for squeaking cracking and if in doubt change belt especially if higher mileage it should be anyway.
The refrigerant itself doesn't actually cool. AC works by using this pressure dependent gas or liquid (hi and lo side) (you can tell as lower diameter line is low side (liquid) and increased, thicker line as a gas. It's ability to transfer heat is why it can "move heat, and in the same way "remove heat." The fluid throughout condenser near fan allows air to sort of blow air (why driving helps it feels colder) over the tiny metal fins of condenser. Attaching high, it gradually turns to liquid cooling with help of fan and fins as well as winding length.
For ac, this means the heater core with fan internally for a gets the low end attachment in order here, rising as it gets from coolest to warmer. Blowing in cab.
The line hits the back passenger corner what people call "the metal thing in back" accumulator. From it the lower diameter (high side as in high pressure) hits an expansion I'll refer to it here as just unit. Then it's easy to see after it follows to you and downward. The drier line really it's course to compressor. The compressor leading to condenser would be low sideor higher diameter low pressure.
So it just keeps blowing out heat this way until it appears to be cooling which it technically is but not magically. For the component nerds out there also loving inner working of the parts.
Anyway
Your rpm should also increase when the compressor kicks on. If not that could indicate a few things being you tuned it.
It would be easier for anyone with ability to go to stock since you can't I would say check rpm settings for anything with ac on ensuring I'd say 100 to 150 rpm higher than ac Off. It give the large kick on of a compressor the "oomph" needed and gets things going. Meaning too if your belt is on its way out it might not get that causing problems.
You need an AC gauge. I suggest Harbor freight. You can get one for probably $20 in pretty decent quality. It is easier to understand attachment as the sides (3 lines middle only if adding or removing refrigerant via a ill just say "refrigerant pump" if you don't know about it don't just buy refrigerant trying to do it honestly if I was only able to replace parts it would be easy to cut price having a tech simply so the refrigerant after. But initially the gauges easily hung on hold help get idea of "at running temp" or other ways of testing.
Not knowing what else may be done I don't want to drive deeper feel free to ask if it's ready now or in future.