Hello everyone i need some help if anyone knows what could be wrong with my car.

I have a 2010 audi a4 and the ac compressor doesnt engage. First i was told by a local mechanic that my control module was bad, so i bought one from ebay and replaced it and had the dealer re program it to my car but my ac keeps blowing hot air. I noticed my ac compressor was not engaging so iasked the dealer and i was told i need a new accompressor for a price of $1,300. I tried to see if i could fix it myself, so i jumped the compressor with a cable directly from the battery and the compressor started working fine and blowing cold air. As soon as i connected the way its suppose to be, then the compressor wont engage. I live in florida and i really need this fix asap. I was thinking about just buying the compressor but im afraid this might not be the problem since it works fine when jumped. All the fuses and relay were checked and they are all good. Does anyone have any ideas if there is another solution? I dont mind replacing the ac compressor, i just dont wanna spend that money and have it be something else. Any suggestion will be of great help. Thank you