Your biggest issue is quite possibly the OEM receiver on your truck. Get under the bumper and you'll see in your era of truck GM relies heavily on the bumper mounting brackets for the weight carrying and weight distribution loads. That receiver design was used from '99 until I believe '08 or '09 when it was re-vamped a little.
What you need is a solid, frame mounted receiver like this
Class V Curt Receiver that takes the bumper out of the food chain. My old 2004 Chevy 2500HD had issues with porpoising and just a slushy feel in when connected to the trailer. Once I installed a frame mounted hitch like the above link it all felt much more solid.
KJ