1996 Chevy 2500: V8 350 5.7L

Questions about my non-functioning A/C.

First, when I run the AC, low or full blast, the compressor does not ever engage.

1. I attempted to refill with r134a, but during the process the compressor still never engaged.

2. I decided to pull the AC relay and bridge it directly to the battery which caused the compressor to engage and run.

2a. While compressor was running I attmepted to refill with r143a, which it took some in. While doing this the AC in the truck was blowing seriously cold air.

2b. When the gauge on the refill can read what I think was good, I pulled the bridge to the AC relay, and compressor stops.. and when running truck w/ AC on full compressor still will not engage on its own (yes, I put the original AC relay back in).

2c. Thinking the relay was bad, I pulled 2 other relays that mathed the same diagram as the AC relay and used those in its place and still compressor does not engage.

What can I try next? Can I rule out a bad compressor and / or a bad clutch at this point? Can I rule out a bad relay?