First thing I would do is determine what an F150, you want or have, can tow. It's going to be more about payload capacity than towing capacity. Unless you get an F150 with the max. tow capacity option, you will find that you may be limited to about 6-7,000 lbs GVW trailer weight. That is, unless you don't care about being overloaded. Some find that even at around 22-24' they are overloaded. If you have factory options, esp. 4x4, super or crew cab and long box, the payload capacity will go down quite a bit. Same with stuff you install, like say a canopy (200-ish lbs). It's best to take your TV through a scale all loaded up as for a camping trip with passengers, pets, groceries, camping gear and a full tank of fuel. You may find that you are under 1,000 lbs available payload cap. Don't go by just the sticker on the door jamb.
Tongue weights vary between 10-15% of the GVW. Unless you know for sure what the actual tongue weight will be, use 15%. Once you have your max. payload, take that number and divide it by 15% to get the max. actual trailer weight you can pull. Say your actual payload capacity is 900 lbs loaded for camping, your max. TT weight would be 6,000 lbs. At 12%, it would be 7500 lbs.
Once you know your max tongue capacity and GVW for the TT, then you can confidently shop for a TT that won't overload your TV.
Don't use dry TT weights. Unless you know for sure what the TT would weigh loaded for camping, use the GVWR. Some units end up close to the GVWR loaded for camping. Listed dry tongue weight can be close to actual or nearly double.
If you do end up being limited to around say 22-24', I think you will find they are nowhere near as functional and comfortable compared to what you have now. You'll want at least one slide (which will add weight). Something with a sofa or two recliners would also be a must for me (adds length though). If you get a 3/4T, you would have a lot more choice and have a better towing experience. After having 4x4, I wouldn't again as it adds height and weight and I never use it.
Beware of anything labeled "ultra-lite" or "1/2 ton towable" because it means they're less substantial overall to save weight and are less durable. Some of the ultra-lites are also using a 3-piece welded fabricated I-beam which is not very strong and can cause issues. Also, just because it says 1/2 ton towable doesn't mean any old 1/2 ton can tow it without being overloaded. I know of one manufacturer that has a "1/2 ton towable" unit that adds around 2K lbs payload to your truck. It's ridiculous and would overload even our 3/4T.
If it's something of better quality you want, I'm reading a lot of good things about Outdoors RV and Northwood. They also make their own HD frames. Lots of TTs out there with a good selection of substandard quality, poor factory support and bad dealer service . Besides trying to find a nice TT, I'd look carefully into dealer quality because a bad one can turn a great RV into a nightmare. Our old dealer was flippin' awful (extremely slow and inept) and our new one is great.