I asked a guy who is very knowledgable about ac components and this is what he said. I know he is long winded, but I hope something in there helps.
"I don't think that the AC was fed through the computer on OBD1 vehicles. That's likely not the problem. Something is wrong with the wiring though.
The power ON for the AC runs through the switches in the dash, then goes through the HP switch and LP switches on the system before it ends up on the coils of the relay. On the relay itself- all standard relays are both normally open, and normally closed- when power is applied to the coils on the relay through all these switches, it closes the contacts on the compressor clutch which is a high current device. That's why the wire is burning up, it can't handle the current from the AC clutch. You should first check the AC fuse, and then the AC diode. You should get electrical continuity through the diode in one direction only. Now, check out the relay diagram and pull the relay so you can match up the pins to the appropriate wires. You should be able to jump the relay out by jumping the constant hot from the big wire to one side of the coils. This bypasses all the safety controls. If it does not work like this, then something is wrong with the relay- an evil coincidence. You can swap this relay out for a headlight relay if that is the case, these should be the same relay.
I suggest this only because it sounds like he's jumping the wrong wires because the wire is burning up. You only need to provide power to the coils, then the relay will do it's job. If it's not doing it's job, then it's the problem. Now if it does it's job, I'd cut it off really quickly after that. Something is amiss somewhere. It's time to test both the HP and LP switches. With the AC off, both these switches should read electrical continuity. If not, then one is likely bad- or the system is horribly over charged, and the HP relay is doing it's job. Yes I know, the system was working before, but that doesn't mean that the orifice tube didn't finally clog itself up right before you swapped things around. Evil coincidences happen like that which is why we test things instead of assuming.
He needs a set of gauges if nothing can be found to be wrong with the relay itself just to be sure that the HP switch is not doing it's job for a reason. The reason I think it's the HP switch is because the compressor is coming on when he doesn't have it jumped out. If the switch was bad- no continuity ever- then the system would not come on at all. If the LP switch was bad, it would not come on at all, and wouldn't come on if the system's pressure was too low. I think what we have here is either a clogged orifice tube, or a possibly bad compressor.
Gauges will tell the truth of the matter. He's at the end of his knowledge with what he's done, so it's time to try something different. He could partially discharge the system and see if he could get it to run for longer, but that would only tell him that there is definitely something wrong mechanically- ie, clogged orifice tube. He could also just tear the system down, flush the components with mineral spirits and compressed air, then recharge with oil and refrigerant after replacing all the O-rings and orifice tube. He'd need gauges for that also.
I would run a lot of tests to be sure what I thought was going on was what was going on. Double check everything- even if you think, oh that can't happen."