Agreed on the 6.0L vs 7.3L. As long as it has been bulletproofed, which is NOT cheap. A few aftermarket suppliers offer bulletproofed parts and they are wicked expensive. For a stock replacement EGR cooler and oil cooler, you're looking at $1000 in parts alone and some pretty good surgery to replace (at least the EGR cooler).
If you take it to a shop and say "bulletproof it" you're looking at an easy $3500. Its one of the reasons I still recommend an LB7 Duramax because even if you have to pay a shop to do injectors its often less money than just an EGR cooler on a 6.0L. And an LB7 injector gives you plenty of warning. An EGR cooler sometimes let's you know it has failed by hyrdrolocking the engine.
Don't go nuts modifying a 6.0L. They only have 4 head bolts per cylinder. Replacing the head gaskets requires removing the cab from the frame.
In the $10k range, I would honestly focus on the nicest Duramax LBZ/LLY you can find. Allison transmission, bulletproof diesel from the factory. Most 6.0L sales are either not yet bulletproofed (expect a large repair bill soon) or they have been bulletproofed with unknown skill and sellers are trying to recoup their investment. That desire to recoup investment battles with their poor reputation for market value and you sometimes have trouble finding a bargain that makes you feel warm and fuzzy.
Just yesterday, actually, I passed on a $3200 6.0L stroke van with 140k because it didn't have the upgrades, but I gave considerable thought to a $7500 LB7 Dmax truck with 150k (and new injectors) but didn't buy it because I'm really looking for a van.
And I will say it again... given their torque curves, ANY of the diesels from the 6.5TD or newer will have more than enough power and torque for towing 10k lbs. I towed a 10k travel trailer with a 6.5 for years and never met a mountain (including the rockies, the Cascades, the Smokies, and the Appalachians) that it couldn't top at or above the speed limit. I would actually have truckers radio me on the CB and ask what I had done to it as I passed them going up mountains. The answer was usually something like "well, I changed the oil last week." The only modification to the truck was the CB radio and a brake controller.
But honestly... despite the fact that I prefer the Ford TRUCK that goes around it, if you gave me $10k to buy a diesel, I would buy an LLY/LBZ Dmax hands down. The only other thing I would consider in that range is a squeaky clean 7.3L with low-ish miles, but then you're paying the novelty tax since there aren't many of those around. I would only look at a 6.0L if the bulletproofing had already been done with receipts, and done by a reputable shop... but then you're paying the "I have something special" tax. I think you'll get more for your money with an LLY/LBZ
If you don't mind gas-like MPG, an LMM is equally bulletproof, but the DPF kills MPG hardcore. I can get nearly 20 MPG in an LB7 empty, but best I have seen in an LMM is 14.
I might also consider a Cummins in that range provided it has the NV5600 transmission. But given their resale value and the fact that most Dodge trucks decay much faster, finding a bargain will be difficult. I would say keep your eyes peeled for one, but don't get your hopes up. The Dodge automatics are on the opposite end of the spectrum from the Allison when comparing the big three truck brands. Expect failures and expect them to be the expensive kind. In my years running transmission shops, I would say the average cost to rebuild (remove, repair, replace) a Ford 4R100 was in the $1800-2400 range. For 47RE Dodges, you can often expect $3500 minimum. The parts themselves are more expensive, and the chances of a hard part failure are almost certain. Dodge focused on engineering clutches and line pressures to support the power but didn't get as much of the hard parts engineering right. What happens is the clutches last forever but they beat the crap out of the steel. Ford and GM focused on bulletproof hard parts, so what usually happens is they last a long time but the clutches wear out and need to be replaced. That second scenario is MUCH cheaper. Its pretty easy to upgrade Ford and GM transmission reliability with easy things: programming and valve bodies. You can't upgrade a Dodge transmission without addressing the weaker guts which requires very expensive upgraded parts and a complete teardown. For that reason, I don't recommend a Cummins with an Auto.
The condensed version: LLY/LBZ will likely get you an easy, repeatable purchase with plenty on the market. 6.0L with upgrades will be equally nice, but the market is all over the place. Cummins with a 6-speed is equally viable, but good luck finding the same amount of quality/miles in the truck part when shopping. Shop for an LLY/LBZ, but don't discount an upgaded 6.0L stroke, a low-mileage grandpa-owned 7.3L, or Cummins/6speed if they happen to turn up in your search