$5000 tow vehicle advise

Crew cab, good shape and $5k is a tough find.
IF you find a 90s 3/4 ton GM 2500 crew cab that is in good shape/low miles, buy it! 90s GMs are still among the best bang for the buck trucks out there but getting rarer. Bonus if you find a 5 speed 350 combo.
Next up, late 90s F150 Crew cabs.
$5k is still a hard sell if you have to have a crew cab.
Best you'll likely find is something in good shape with a ton of miles on it and 20+ years old.

A 2000 or newer crew cab capable of pulling 7,000 pounds for $5,000 is a tall order. It will either have tons of miles, be beat to pieces, or both.
I would look for a GM with the 8.1L engine and Allison trans. However, for the budget you will probably be stuck with a 6.0 engine. It's still a very good engine but doesn't have the power of the 8.1.
Dodge trucks from the late '90s and into the early 2000's had problematic automatic transmissions. I had one that had to be rebuilt at 90,000 miles. Even when working properly I didn't like it. It never seemed to be in the right gear for the job. I manually turned OD on and off and shifted between 2nd and 3rd manually; basically driving like a 3 speed on the tree. If you get a Dodge I'd get a manual for sure.

You will be safe with about any possible optioned Class 3 crew cab. Look for the best value on a F350/3500 SRW model - Any of them, regardless of power train, will have the capacity to tow your desired trailer.

Once you get a trailer, get an actual tongue weight with it loaded and make sure the receiver has enough capacity. The rest of the truck will be fine.

The trailer would be ~6500lbs empty, bumper pull. Truck would be from around year 2000 and has to be crew cab. Can't buy the trailer first, would have no way to pick it up. Do not plan to do any mountain towing

So any pointers on model year 2000 or so gas trucks that can tow 7000+ lbs? I.E. stay away from this engine or this tranny or this year Ford etc.

You really need to pick out a specific trailer/fifth wheel first. Then you need to decide if you are going to be towing in the mountains or not, that will tell you whether you need a diesel or not. Of course if you get a light weight bumper pull then a gasser will be fine. Another reason to do the trailer first.

Getting a 3/4 ton vehicle in any kind of shape for $5K is going to be difficult, but not impossible.

We found a 7 yr old Dodge 3/4 ton, 2WD, 5.9L, tow package (tranny cooler, HD battery & alternator, HD springs, class 4 hitch) fairly new rubber, king cab and 8 foot bed with no rust and 80K miles on it for $5,600. We have been driving it now for 5 years and she still runs great and pulls our trailer very well.

Good luck on your hunt. We'll be in the hunt again in a couple of years, I would guess.

Wow! That's like asking what color of paint should I paint my bedroom. The sky is the limit.

Posters on these forums seem to say, select the camper first, then get the truck to match. For the longest time I disagreed with that statement. It's only taken me 60 years to finally figure it out ... (they ARE right). Find the trailer that interests you, then begin looking for the truck that will provide a comfortable tow. Otherwise, the only trailer you may find you are able to tow will be a pop-up. By the way, a 3/4 ton WILL handle about 90% of the campers out there. But you also have to figure out gear ratios and hitch weights too.