Using 2015 F-350 SRW diesel as the example. GCWR is 23,500, tag-trailer tow rating is 14,000, and GVWR is 11,500. A 12.000 pound tag trailer with 13% hitch weight will have 1,560 pounds of hitch weight. 1,560 pounds hitch weight leaves 9,940 pounds max truck weight before you exceed the GVWR (and payload capacity) of the truck.
12.000 pounds gross trailer weight plus up to 9,940 pounds wet and loaded truck weight is 21,940, or well below the 23,500 GCWR. Even the heaviest configuration of F-350 SRW probably won't gross more than 9.500 pounds before you tie onto the trailer - unless you load the truck down with a bed load of green oak firewood.
The GCWR indicates the max weight your rig can have without overheating anything in the drivetrain and without being the slowpoke holding up traffic when climbing hills and mountain passes.
So based on the numbers, with a 2015 F-350 SRW diesel, you could easily tow a 12,000 pound TT comfortably, with no white-knuckle driving. That conclusion assumes you have enough brains to hook up the trailer with a ProPride hitch, or at least one of the $1,000 no-sway hitches such as Reese Strait-Line. Equal-I-Zer, or Blue Ox SwayPro. Any new weight-distributing hitch you can buy for less than $600 (complete with adjustable shank) is a cheap hitch that will probably result in some white-knuckle driving.
Here's my ProPride:
Trailer Sway Elimination | Sway Control ProPride, Inc.
And here's my Strait-Line, but with your spring bars rated for 1,500 pounds max tongue weight, which is the minimum you need for a 12k trailer.
Strait-Line Weight Distribution w Sway Control - Trunnion Bar - 1,500 lbs TW. RP66130
Note that one is less than $600, but that's because it doesn't include the adjustable shank. So add the adjustable shank:
Reese Weight Dist Shank - 12-1/4" Long - 2" Drop to 6-1/2" Rise - 1,500 lbs TW, RP54970
and your total is more than $600.
The Blue Ox SwayPro and the Equal-I-Zer will also cost more than $600, but that's a lot less than my $2,500 ProPride.