On the 6-speed autos, the Tow/Haul changes shift points, downshifts through the gears with the exhaust brake, and might do more transparent things. On the 4-speeds does it lock out overdrive like it did on my 46re in my Dakota?
At 7500 pounds, I'd consider running it 24/7 whenever the trailer is hooked up. In my Dakota I did run with tow/haul off on the highway while towing a 5k pound trailer, but flipped it on whenever I hit any hills at all.
As far as the gooseneck is concerned, according to dodge.com, the trailer you're looking at empty is about the max your truck can carry.
It's stupid, we both know your truck can haul a 30k pound trailer without struggling, however the door sticker has your legal limits.
Do you have 4.10 or 3.73 gears?
3.73 = 9900 gross, 3005 payload, 9950 max trailer weight.
4.10 = 9900 gross, 3005 payload, 11950 max trailer.
Those are generic weights though, your options will affect that. Pin weight is going to count towards your payload, as will you, your wife, and anything else you put in your truck.
Max combined weight of your truck + trailer is either 17,000 pounds or 19,000 pounds, depending on gearing.
You need more truck.