Any time you break open a refrigerant system to replace a component, you can't just button it up and put in new refrigerant. You have to make sure that the lubricant charge is correct for the system volume, and you have to pull a vacuum on the system to remove moisture and air so the system is free of contaminants and can be refilled properly. The vacuum pull is also to leak-check your work, if the system won't hold vacuum, it won't hold refrigerant either. You'll spend far more than "the 2nd $100" on a vacuum pump, gauges and a cylinder of R134a.
I do all my own car service and maintenance, but A/C service is the one thing I leave to the pros who have the right equipment. If they've offered to change your compressor for $120, you might want to jump on that, especially if you want working A/C after all is said and done.
1985 SVO Mustang (turbocharged track whore), 1989 Taurus SHO (supercharged track whore), 1999 SVT Contour (Sedanus-Grocerygetter-Rapidus), 2008 Mercury Milan Premier (Comfy boat).