Better general definitions by the OP are needed in order to plumb the depths around here. Better specifics are requested as one can go in any of a number of directions based on what is written (thus far).
So, for what is available:
I wouldn't bother with a 3/4T and would just go ahead and put the one ton trucks on the list. For what you are doing an 8' bed on a real pickup is easier to live with (to continue what BartS is trying to say). They both ride about the same unloaded, but one has better payload capacity. (Ride "quality" can be changed with progressive spring packs, shocks, etc).
A bed topper gives you another 170-cubic feet of storage and no worries about weight. I don't recommend these trucks as the "obvious solution" to TT towing (there are other, sometimes better choices), but a truck is really
hard to beat until one knows what to pack and what not to pack. And that takes a good deal of experience. Full-timing means 4-season clothes and supplies. May want to address whether you'll be returning to a storage facility to change out supplies based on climate/terrain to be encountered.
The "thinking flaw" of someone new is in underestimating the need for space and weight of supplies required. They tend to multiply exponentially. I don't in any way respect
those who tell me that they themselves are simple in needs or wants. This Conestoga wagon has needs of it's own, and that
to be respected. After time, one gets better. But full-timing with no home base is the deciding factor.
On tools -- alone -- the weight/space requirement is high. The "flaw" is continued in thinking that what
I am doing is primary to how
I am doing it when the order is actually reversed as the means dictate the ends.
The Tundra wouldn't even be on my list. Nor would a diesel Ford. Nor would 4WD. Take some time and identify brand, year and model and truck spec more closely according to budget (Auto or manual trans, etc). There is a good range of experience here that can take into account all
those differences. $12k buys one truck, $27k buys another where totally different brand and spec according to what is "best" for the odometer mileage specified. (Which is rather low according to what trucks are capable). Be a bit more specific. You'll find someone who has -- or has had -- that TV.
I'd also recommend EDMUNDS True Cost to Own
as a departure point to understand cost-per-mile calculations. Some truck spec's mean a $1/mile (and higher) op-cost. Mine is 40-cpm. The difference in one year
over 20k miles between them could be $15,000 in adjusted expense. This may matter or not. It does to me: were I moving the trailer 300-miles weekly that would be at 24-cpm, and solo miles would be 16-cpm fuel cost (a high, conservative estimate for my rig; a gasoline truck would be double this cpm). Overall
economy is the bigger picture, thus some trucks will actually fit what you want better than others by this analysis tool. And, as none of them are cheap to own/operate, the trade-offs are crucial (and parking them at the grocery or "ride quality" are actually a ways down the list of what is important).
Good to see you already have a sway-eliminating hitch, so add trailer disc brakes and a state-of-the art brake controller as well (Brakesmart or Direclink). There is no adequate substitute, especially with high annual miles for these items. Read 2Air
's posts/threads on hitch rigging, brakes and tire maintenance. Numbers, numbers, numbers via certified weigh scales (hitch rigging) and setting up controller for surfaces (paved, unpaved, wet, etc; know
the brake boost settings) and how to "read" tires and interpret results.
With that age trailer (1999) I'd also be concerned about axle condition (plus alignment).
16" wheels and LT tires would also be on my list (and with a diesel truck, an exhaust brake unless factory-equipped) as well as Load Range E MICHELIN (LTX A/S or M/S) or BRIDGESTONE (Duravis 500 or 700) tires only
on that truck.