Shopping for a different tow vehicle.

For the most part the 7.3 is a good engine. The rest of the truck I would worry about front end ,trans, brakes and bearings

I don't think you're going to be much happier with that old F350 if your problem with the current truck is power.

What exactly is the issue with the current truck that makes it pull "like garbage?" Maybe we can address that.

Where did you get the info that said your truck was rated to tow 10,000? My 2015 Silverado daily driver is only rated to tow 6200 per the owner's manual.

Tow rating depends on engine, transmission, and axle ratio. Mine came with the 3.08 axle ratio, and it seems like most trucks had that ratio. To tow 10,000 with this year truck you need the 3.73 axle ratio.

I have the 3.73 rear end with 5.3L V8 I have trouble when on the highway with wind affect. I have a weight distribution hitch set up by the book and two sway bars installed

Possibly you need a heavier truck.

That's why I'm considering the one ton truck it's longer and built. It's got the 7.3 turbo diesel with a Manual transmission

I wouldn't care to back a trailer with a manual tranny, but if that suits you, ok with me.

If there were a system to install on my Silverado I would do it. I just can't pin down a system that would take care of my issues I'm having. I thought maybe a air suspension would stiffen up the truck

Put a few hundred pounds (maybe water) in the bed and see if that helps.

Stout pretty bulletproof truck. Apples to apples, the size and suspension will tow better from a handling standpoint.
You said built. Could be good, bad or mean nothing at all, if you meant performance or engine work.
Depends on the condition of the truck and how handy you are and how willing to repair/keep up a 20 year old truck with 200kmi.

Invest in a Hensley Arrow or Pro Pride hitch. It'll tow like a dream. You should have plenty of power for 6000lbs. Lots cheaper than buying a 24 year old truck.