A/C stopped blowing cold, not sure compressor is running at all

So strip off the connector and hard wire into the clutch? That's another option.

We're gonna first attempt a better solder job with heat shrink and interweaving the two wire ends together. I also noticed a mounting clip got brittle and broke, and that's what gave the cable so much play to where the wire broke loose. So I'm gonna get that properly mounted again. If that ends up breaking loose again, I think I'll get the connector from a junkyard with as much wire as possible attached, and splice it at the same point it normally splices onto the main harness. I'd rather avoid having to hard wire anything since if that ends up breaking loose, I could need to replace the clutch assembly.

Alright this is fixed. Watched some videos on better soldering techniques. We flayed out the wires and twisted them together. Then applied rosin flux, heated up the wire from underneath and applied the solder above the wire (60/40 lead/tin Rosin core). Then added some dielectric grease and heat shrink tubing, and heated that up to make a good seal. I then used some bailing wire to mount the wire's shielding back onto its clip, so it has no play anymore (this is over heavy plastic shielding so no chance of shorting out the wire). It seems the clip plastic is now brittle and broke free, causing the connector to have too much play, and broke loose due to vibrations rattling the tiny wires.

Unfortunate update on this. A/C just stopped blowing cold while running it full blast on a really hot day today. I assumed the soldered wire came loose again, but it is still attached. So this is a new failure in the A/C system. When I turn on the A/C, it does not appear the A/C compressor is spinning at all. I do notice that the IAC is still adding RPM's when I turn the A/C on, and reducing RPMs when I turn A/C off. I've had an A/C compressor go bad in a different car once, and it made all sorts of horrible noise. This is totally silent, aside from a little buzzing sound seeming to come from the blower motor. It's blowing just fine though, just hot and not cooling at all. Any ideas?

The low side is 37 psi. It was 34 psi before I added a bit. I don't have a high side gauge. But I'm pretty certain it's not the refrigerant, as it was working perfect one second then blowing hot the next. Not the type of behavior I'd expect from a gradual leak of refrigerant. I would expect a gradual decrease in cooling ability over days or weeks.

This was the same issue as before. Our solder point near the plug had wiggled/melted loose again, taking the rest of the wire off of the plug. So we went to the junk yard and got a replacement cable with plug. Then we spliced it much higher up where it was nowhere near the plug. Heat-shrink it, wrapped it in electrical tape, replaced the shielding. Now it works perfectly again. This was much more of a permanent fix this time.