connected the signal wire should have voltage with a charge in the system. you need to test with a voltmeter

 

I believe the 3 pin round connector in the small line near the compressor is the high side pressure transducer switch. I had to replace mine in my 06 Burb.
Have you tried jumping the compressor cycling switch on the accumulator, passenger side firewall.
It is a 2 pin connector. Try jumping the connection with a paper clip and see if the compressor engages.