Best regarded brands - Generally speaking

The height requirement drives you towards a 5th wheel. You can certainly do a travel trailer but the roof will always feel like it's right there about to hit your head.

Assuming you get a mid-height 5th wheel, you get a lot more head room everywhere except the bedroom but in the 26-30' range, you don't really spend a lot of time standing in the bedroom anyway (if you are comfortable crawling in and out of tents, this should be a non-issue).

Are you comfortable with your feet hanging off the bed a bit? A lot of RV beds are 6-8" shorter than a household bed, so if it's critical you will want to look for a full length bed or plan to replace with a full length.

With a 5th wheel, a 3/4 or 1 ton is really preferable as the pin weight is usually the limiting factor but if you stay in this size range single rear wheel should be fine.

Assuming you go used, past maintenance tells you far more than manufacturer. Most of the equipment and construction are largely the same unless you go really high end but then you are going to find it tough to get something reasonably new in your budget and it's going to be heavy (possibly enough to drive you to a dual rear wheel).

PS: I wouldn't consider not having at least one decent sized slide in the living area. The difference in living space (particularly with a couple kids on a rainy day) is just night and day. Also, the resale value will be hurt as almost no one wants a non-slide trailer.

Go to a dealer and sit down and imagine functioning with normal daily tasks (particularly if it's rainy and everyone is inside). With your height, some of the small toilet spaces are an issue. It sounds crude but pretend you are wiping. Does your head hit the wall during the process?

I would say 1/2 ton TV stay under 7500 gross TT weight, 3/4 if you want bigger.

With kids, a slide is a good idea. Floor plan remains when all else is forgotten.

Also, found the TT we thought we wanted and rented a similar sized unit for a week. You learn quite a lot that way.

Ditto: Nash (laminated wood frame wall structure), and Outdoors RV (laminated aluminum frame wall structure.)

Outdoors RV (and some Nash) has HD frame and "off-road" chassis, along with a high crown radius roof, which even after the 5" insulated crown, still markets more headroom than the competition. You'll likely have to head out West to get one, but many folks do just that.

We don't need 4 seasons construction in our neck of the woods, but I value it for strength and capability in dealing with our heat/humidity, and the benefits of the stronger frame. They are heavy, so pay attention to your loaded weight.

Here's a clip of their construction, tells a good story.
Outdoors RV construction video

Here's an "out of the box" option. Reading your post sounded familiar. (tent, interior height, A/C, 1/2 ton truck, hunting, 2 boys, and you are handy). We went from two tents and a loaded family van to a cargo trailer conversion. It has to be the most fun camping project for the lowest cost we've ever done.

Here's our basic toy hauler/camper, just one step up from the tents, but the sky, and your imagination is the limit:
cargo trailer conversion

We ordered ours with windows, RV door, jacks and step installed, along with 6" additional roof height from the factory, and we finished out the interior, added electrical, insulation, and the modular e-track system that allows us to quickly convert from camper, to hauler, to plain old trailer. You could easily install a roof A/C, but we chose to build a removable 1/2 door width "closet" that holds a window A/C unit.

While it looks fairly utilitarian, we really like the welded 1" steel square tubing on 16" centers in the walls and ceiling. It really makes for solid construction, and I don't worry about taking it down a forest service road for example.

Here's another discussion from a new builder and some helpful links to give a better idea of the conversion concept.
Cargo Conversion ideas, family of 4

Most of us here, will find the camper we like, and purchase it from a dealer ready to go. Based on your list of requirements, an Outdoors RV or Nash would still get my vote. Want to save some money on the tow rig, have a fun family project, and put in the camper only what you really need, and have cash left to enjoy the camping experience without making payments? Then consider the cargo conversion.

Edit: Speaking of finding the right "turn-key" camper. This one is all aluminum, built tough, sleeps 4, ready to go, (on sale)
Livin' Lite Camplite 21RBS

A walk-thru video on the same model:
21RBS walk-thru

Good Luck

Are you going to stay on pavement, or tow down dirt and gravel roads?
If you are towing down dirt and gravel roads, look at Northwood RV, Outdoors RV, and Lance.

If your handy on the PM work, you know much of what to look for.Appliances are similar in most, frames and axles in general are similar and designed to the level stated on the manufacturer description, they don't tend to overbuild. Interior fixtures cabinets , you know the drill, screws are better than Staples, wood even plywood is better than particle board. Fit and finish will tell you much.
Warranties tend to be similar, a few offer 2 years , Jayco is one that does.
The 6'6" is going to be tight, most have that height, but I'll bet you tend touch light fixtures and AC units. A 5er may do ya a little better there.

Jayco must be doing something right being the most popular brand.

If you could find a Nash or Arctic Fox would be my top picks.

Keystone Premier, Lance and Grand Design trailers are the first that popped into my head. I'm not sure about interior heights of them since I'm only 6' tall :-) but for all the other features they'll work.