You have MANY, MANY options. You do need to shop with the weight in mind. But, DO NOT eliminate any length of trailer because it looks like it will be too heavy. New TTs come with a "yellow" sticker. On that sticker you will find the weight of the TT as it sits on the lot in front of you, with every thing you see on, or in the TT. It will also tell you the weight with a battery, and propane, plus the weight with full water tanks. Really good information to have. It will also have the carrying capacity on it. So you will know how much it will carry, and what it will weigh loaded to the max. That can be use full, or meaningless, depending on how you camp. I will NEVER load my TT to it's max, so it means nothing to me. Some are married to Lucy, and they will load theirs to max, so it mans something to them. how you load your's is totally up to you. Just be aware of all your numbers.
As to length. Means nothing as to how much it weighs. My TT is 31 feet long has a 18 foot slide, and weighed 4800lb dry, meaning as it sat on the lot. After I loaded all my camping stuff in it. It weighs a little less than 6000lb, and that is ready to camp. With everything we need to have a great trip. Your TV would have little problem towing my TT. Would do great.
You would need a WDH. A properly setup WDH. The key to towing with ease is getting the TT, and hitch adjusted correctly. IF you get it right, you will have no sway. Best thing is to get it right, and then, and only then add sway control. Never try to fix sway with a sway control bandaid. Fix the sway first, no trailer should sway if it is properly setup.
And it would help to get a heavier tire for the TV. Most trucks come with tires rated to 2271lbs. they do just OK. However a tire rated to 2601lb, and 50 psi will handle the TT much better, and always keep the trailer tires aired to their max, usually 50 psi.
Most important. Have fun, and be care full. The TV with the TT will take longer to get up to highway speed, and will take much longer to stop, so leave more room and start stopping sooner. It will also take more room to turn, so drive accordingly. And enjoy. It is not rocket science, it is not hard. Or every one couldn't do it. Just pay attention to detail.