The biggest problem most newbees have is underestimating the weight and pin weight of a trailer, so they wind up overloaded on the road. Don't do that.
I'll assume a Mobile Suites 40 KSSB4 as your full-timer choice. GVWR 21,000 pounds with 22.4% pin weight = 4,704 pounds pin weight. Full-timers haul everything they own with them when they move the trailer, so don't assume less trailer weight than 21,000 pounds or less pin weight than 4,700 pounds.
The 21,000 pounds gross wet and loaded trailer weight is not a problem. The Ford F-450 pickup can tow that much and more without a sweat. But the F-450 pickup cannot carry the 4,700 pounds of pin weight, plus the weight of a good 5er hitch, plus the people and stuff that will be in the truck when moving locations. Remember, you must haul everything you own when you move locations.
That late-model F-450 pickup, 4x4 CrewCab, has a maximum cargo weight with slide-in camper rating of 4643. Almost enough for the pin weight, but with nothing else in the truck. So not quite enough truck for that big 5er.
So forget abut a pickup. Nobody makes one that won't be overloaded with that 5er when wet and loaded for a full-timer move.
So resign yourself to beginning with a chassis-cab truck with enough GCWR to pull 21000 pounds plus enough GVWR to not be overloaded when loaded down with 4,700 pounds of pin weight plus the other weight in the truck when moving. You can buy an F-550 chassis cab truck with Lariat trim, then have an upfitter add a tow body. The F-550 is available as a regular cab, SuperCab or CrewCab, and 4x4 or 4x2, and gas or diesel engine. Forget the gas engine - you need the PowerStroke diesel to pull that monster-size trailer over hill and dale.
Here is a suitable tow vehicle I built and priced on Ford.com.
2017 Ford Chassis Cab - Build & Price
Sticker: $67,760 Total MSRP
Model
$63,520 2017 Ford F-550 Chassis Cab LARIAT Crew Cab®, 179” Wheelbase, 7L 4 Valve OHV Power Stroke® V8 Turbo Diesel B20 Engine, 6 Speed Automatic with SelectShift® Transmission (6R140), 4X4, 4.30 Limited Slip Axle, DRW
$1,080 Trailer Tow Package - High Capacity
$125 Extra Heavy-Service Suspension Package
$710 LARIAT Value Package
$125 28 Gallon Mid-Ship Fuel Tank (in addition to 40-gallon main tank)
$350 Spare Tire, Wheel and Jack
$0 PowerScope® Telescoping Trailer Tow Mirrors
$370 Platform Running Boards
$0 19.5" Polished Forged Aluminum Wheels with Bright Hub Cover/Center Ornament (DRW)
$0 225/70Rx19.5G BSW A/S Tires
$0 Leather
$0 Trailer Brake Controller
$0 Electronic-Shift-On-the-Fly (ESOF) - 4X4
$0 110V/400W Outlet
$570 Voice-Activated Navigation
$0 SYNC® 3
$0 Premium Leather Seating Surfaces, 40/Console/40 Seats
Okay, that’s the fancy, powerful, heavy-duty tow vehicle. Tell the dealer to “ship thru” an outfitter that will add the tow body with 5er hitch. That can add a few thousand to the total price, depending on which bed and hitch you choose. “Ship thru” means the factory will ship the chassis cab to an upfitter that will add the bed and hitch and then ship it on to your dealer, with no additional shipping charges to you.
There are a few Ford Truck-only dealers that might have your F-550 in stock. One is in Orlando FL. But probably not. So plan on ordering it and waiting about two months for delivery.
You don't want a Ford? Then you can buy a similar medium duty (class 5 or 6) chassis cab truck with single dually axle from other brands, including Peterbilt, Freightliner, GM, Ram, and others, and have the tow body added by your favorite upfitter. Or if you're good at searching the internet, you might find a used one that won't require a two-month wait for it to be built and delivered to you.
Instead of a tow body, you can add a pickup bed. But the high sides of a pickup bed means you'll probably have to lift the DRV to have enough clearance over the bed rails to make turns with the 5er. So I would stick to a flat bed or a tow body. Or maybe a Knapheide Westerner bed:
https://www.knapheide.com/gooseneck-...er-storage-p21