The housing rental market has been insane for the last couple of years. That may (I hope!) be peaking now, but it's still scary as h3ll, and some things (like corporate ownership of housing stock) will probably never go back to "normal." So when you say
lay down for a bit while we get a new rig together.
it's worth asking yourself: how long do you expect that to be? and is it worth upping stakes, purely to find cheap housing, for that length of time? The answer might be "yes," but think it through. Travel to a new housing market has its own expenses and risks.
Speaking as a serial renter who has covered 2000 miles in the last 16 months (and God knows how many in the last 20 years), here are a few carrots to throw in the soup pot:
It might be worth looking at some of those "10 cheapest cities" lists -- which you can find dozens of online -- but don't take them too seriously. Like any other "10 best X" lists, (1) the main ingredient is often fantasy, and (2) if you found it that easily, so will a zillion other people.
As far as I can tell, newspaper ads are pretty useless for rentals anymore. The main sources seem to be Craigslist and various real estate sites like Zillow and Rent.com. Craigslist is infested with scams, and the real estate sites are often out of date or run on some stupid algorithm that doesn't work right, so you have to be careful and plow through a lot of junk to get to the gold.
If you have ANY contact you can exploit for housing clues -- your sister's wife's hairdresser's boring third cousin used to be a plumber there -- do reach out.
A rental market might be expensive or cheap
on average, but you only need one place, and your personal luck and timing will play a big role. So don't get too complacent, but also don't get too easily scared off from someplace you really really want to be.
In my opinion, your odds are better renting from a small landlord, if you can find one, than from a large property-management company.
Possibly crazy but I'll throw it out there: If house-hunting in a new area -- especially if it's not a big, homogenized city -- consider getting a burner phone with the local area code to make your house-hunting calls. In one place I lived, I had rental agents jerking me around like we were 15 years old and they were hoping someone better would still invite them to the junior prom, and I speculate that it's because I was not from there. You'd think the whole world would be acclimated to cell phones by now, but no.
You're not on the tightest of all budgets. You might want to look into extended-stay motels, like Motel 6/Studio 6 and whatever Value Place turned into (Woodspring? Windspring?). They're more expensive (and imo less nice) than they used to be, but if you can afford them, they give you the most flexibility as you're only committing for a week at a time. If you go this route, check the reviews -- but also check for advice on how to evaluate the reviews. There are plenty of bogus reviews, but there are also good tips on how to weed out the bad 'uns.
Random vicarious fantasy thought: when I was planning my next move, I thought about Tulsa. Cost of living looked relatively affordable, and it looked like a pretty cool city, especially if you like outdoors things. Worth a look, maybe?
^^Those are about the mechanics of it. From a bigger-picture angle -- and I can't stress this enough, because I would love for other people to be able to avoid some of the face-plants I've made -- PRIORITIZE. You're working on three things right now -- getting a rig together, changing careers, and changing locations. Each alone would be huge. You probably can't do all three at once. It will feel like you can, but that feeling can be misleading. MAKE A PLAN to tackle these things one at a time -- or at least to control how much time and energy you allot to each, if you do them simultaneously. Any time you start to get that "yeah, I could probably have my cake and eat it too here, let's just wing it for awhile" feeling, stop and think it through. There's a reason they say "one step at a time."
If you don't have a ton of stuff (I'm guessing, since you live in a tiny house already, you don't), I'm sure you can find something for <$1600.
Good luck!