05 Silverado AC compressor cycles on/off

Hi.
I thought I saw every video how to accomplish this rather seemingly painless procedure for Recharging AC on my 2005 Silverado 5.3L.
The clutch engages for 4-5 seconds and cycles off. Repeats but never stays engaged. Refrigerant loaded to proper level via meter. No obvious leaks. Realizing the importance of maintaining the correct pressure, I noticed the gauge jumped when clutch engaged , then dropped out when cycled off. The cycle sensor switch looked clean. My auto part guy thought the same. The fuses looked good. THE only thing i see odd is my cooling fans didn’t come on even after 10 min. I realize the thermostat could be stuck open. Would that also affect the AC system?? Thx in advance.