2001 Dodge Stratus Compressor Shuts Off

Greetings! I have an issue with the A/C in my Stratus; in the mornings it works fine. If it sits in the sun all day and bakes (in AZ that's what cars do!), The A/C compressor will kick in fine, but after a few minutes of driving it shuts off and car blows hot air. The car drives like the A/C is not on. I noticed that several months ago the dash lights on the climate control module stopped working, I thought this might have been the problem. I also noticed that I had to hold down one of the knobs on the module to get it to kick in; that only worked for a while. I also noticed that on the back of the module, with the air running, it got EXTREMELY hot where the large harness connects to it. I found a replacement module on ebay and installed it. No dice. I then thought the A/C relay might have been bad, so I replaced it. No dice. I checked the pressures with a DIY A/C recharged kit that I had (b/c a long time ago the A/C was blowing hot and it turned out it was very low on refrigerant) and it indicates sufficient charge. Any ideas? If I pull the climate control module off the dash while driving the A/C kicks in (sometimes), which makes me think possibly a wiring issue? I don't want to take it in to the shop to be told that I need to plunk down $1K for a new compressor, etc, when I think the mechanical components of the system are working. Thanks!