I saw this:
Am I missing something? How else am I supposed to get a job? I don't even hear back about why they pass me over.
...and I was thinking, gosh - he's throwing up his hands and giving up without really trying.
But then I saw this:
[...] This is a small company and I'm the only programmer. [...] I mostly do everything myself. Some tools I commonly use are HTML, CSS, JS, PHP, Python, Ruby, Java, SQL, Mongo, and Git. That's without listing their respective libraries. When I have downtime at work, I work on either my Emacs config or my own websites. [...]
and started thinking "how the heck is his company getting away with paying him that little?"
It sounds like you have mastered (or are mastering) a large set of technologies, are self-directed, and can get things done. These are surprisingly unusual skills.
Back to your original question, OP:
I have problems with winter depression, so I want to go somewhere with a warm climate.
[...]
What would be a good place to move to in the USA?
I'll give you differing advice from most of this thread, given your skillset. You don't need to move to a LCOL or a small city with low property taxes. On the contrary, you're the perfect candidate to move to one of the most HCOL, impossible-to-buy-real estate areas in the country: The San Francisco Bay Area.
Starting salaries for Software Engineers fresh out of school here are six digits. Rents are commensurately ridiculous, but if you're willing to live minimalistically, even though you'll be saving a smaller percentage of your income, it'll be a bigger number (probably by far).
The weather can't be beaten: Not too hot, not too cold, and rarely rainy. (Beware of fog along the coast and in significant parts of San Francisco itself).
How to do it:
Are you on LinkedIn? If not, get on it now.
I wouldn't normally recommend this path, but you may want to see if you can find any recruiters who can help you find a position - they will most likely also offer you some advice on how to structure your resume to work better (for one, make sure Software Engineer or something like it is in your job description).
Do you have a college degree? I'm guessing you may not and that's why you're having some issues. Don't despair: the market is hot enough that this is not an insurmountable problem. You may have an easier entry into the market in a non-developer role (though that may also make the switch into a developer role more difficult down the road).
With you self-described skills you should eventually get involved in an early-stage startup.
Regardless, Silicon Valley is *the* place to be for Software Engineers. Job mobility is tremendous (once you establish a bit of a job history, that is), so if you don't like a position - find a new one.
The main thing that worries me about this scenario is:
My goal is to be able to stop working as soon as possible.
I'm thinking that once you're in an environment with other like-minded people, you may find that the work is an activity that's worthwhile in itself (in addition to having the potential to give you a large amount of money). You'll still be able to retire early, though perhaps not until you're closer to 40.
If you really just want to stop working, then disregard my advice.