I have noticed a slight vibration when pulling away from a complete stop. This when pulling my 6,000lb trailer and using Tow/Haul mode. It's not noticeable without the trailer.

Without the trailer, the truck does not pull away smoothly. It lurches slightly, and you hear/feel the thud from the rear axle. You have to be really light on the throttle to avoid this.

Are these signs of a torque converter or transmission problem?

 

Check your driveline for possible bad u-joints or carrier bearing.

 

Have you ever serviced the trany, fluid change and adjusted the bands, I do mine every 30k miles. 137k and still shifts fine.

 

Yes. Trans. oil has been replaced a few times. Had a problem early on with constant shifting 1-2-1 at low speed, fixed under warranty. I don't believe the bands have been adjusted. But the truck only has 68k