jumping power to the green wire is a test, just to see if it runs. you dont want to keep it like that. you see it run, you take the power back off asap.
sadly sometimes a/c systems are best left to the pros. and i know they like to charge out the ass. but they deal with it everyday and have the right tools and equipment to get to the root of the problem.
and they wont waste your r-120 whatever as they will suck it out and tank it, then reuse it after they fix the system. where if you open the system your just dumping it into the air then have to buy new. unless you have these cool tools they have.