So, it blows without the AC relay jumpered? If so, your problem is wiring somewhere under the hood, probably to the DCCV or at the front right corner.
Don't jumper the AC relay for more than a few seconds. Otherwise, you could be doing severe damage to the whole AC system. The clutch isn't engaging because the PCM is disabling it to prevent damage. The problem is almost certainly low refrigerant. I say this because of your charge story. The AC doesn't consume refrigerant. If it was low, it was due to a leak. You need to find and fix the short, and you need to find and fix the leak. Just throwing parts (compressor) at it is unlikely to help at all.
Common leak points that I have seen:
O-ring for the scroll control valve. This is on the compressor, so replacing the compressor would fix it, but you don't need to. It's a $10 part and you can replace it with the compressor still on the car.
Low and high side port valves. These are also $10 or so parts.
Of course, anywhere there are connections, there could be a leak. The factory did put a UV dye in. You could check for the dye everywhere (need a UV light and glasses), clean it off everywhere you find it, top off the refrigerant enough for it to run, then check and see where the dye comes back.
Once you find it. You'll have to discharge the system. Correct the leak. Replace the dryer. Add oil as needed. Pull a good vacuum, and then refill the system. Note that those last three steps are tricky. You really need to know what you are doing, and to have the proper tools.