I am in TX with a budget of ~1500 for a week or two. The longer and cheaper alone time is better. What do I like, you know I've been so slammed and in my head that I can't even pick a vacation spot or remember what I like. I think I'd enjoy just some peace somewhere, disconnected from it all for a week or two.
My biggest suggestion would be to get out in nature. Unless you are really opposed to spending a decently long time without creature comforts then I think this is one of the best things.
Another piece of advice is to stay away from crowds like you pointed out. There is a seemingly infinite amount of destinations that could satisfy this requirement too.
Best of luck! I hope you find something suited to you and feel refreshed after getting back.
There are few things that work better than outdoors sports for me, so I agree on that one.
You might be interested in the concept of masculine / feminine societies (checkout geert-hofstede.com), because as a texan, you are probably in one of the most masculine places in the world, which is probably rather stressful. I would suggest, if you decide to travel abroad, to go to a country that is considered to be a feminine society, like Sweden or Thailand (avoid the south though, and the party islands). I was astounded that a huge city like Bangkok could feel so relaxed, it is a different life.
Though I have to say, traveling is often not advisable when you are depressed. You have higher expectations of happiness that are impossible to fulfill, plus often higher stress levels, which will likely make matters worse.
See if you can rent out a lakehouse in the Finger Lakes of Upstate New York. Not much noise, few distractions, plenty of nature, plenty of wineries. Kick back with a bottle or two and listen to the birds.
Get a backpack, a tent, some supplies and go hiking. It costs next to nothing, can be done almost anywhere, gives mind clearing exercise, and gives you concrete and solvable problems to focus on.
Perhaps best of all, you'll be outside of cell coverage.
What about going work in a farm (something like fruit picking) to be sure to be busy with something different, and to feel physically (instead of mentally) exhausted at the end of your day.