I have come across a couple of them while identifying aerial photos. I have only found this term in Arkansas, Alabama, Louisiana, Mississippi and Texas.
Just another name for a laundromat.
Sounds like a combination laundromat and cafeteria - not a bad idea, from a customer standpoint. (Whether it works as a business model, given the diverse stuff that would need to be managed, I can’t say.)
For a few years the laundromat I used was attached to a… bar - probably a bad idea.
There was a franchise concept for a while in the '80s: Duds & Suds, which were a combo laundromat/bar/arcade.
That’s where I usually did my laundry, when I was in college in the early 1990s.
This.
No food or drink service implied.
This article indicates that Washateria was an early self-service laundry chain, in Texas and parts of the South, but the name was supplanted in most of the US by “laundromat” (which was itself originally a brand name for a Westinghouse washing machine).
I honed my Galaga skills at one of those.
I grew up with washaterias in Texas (but then our local wrestling arena was called The Sportatorium, so we had all sorts of fancy names for places).
I only heard the term “washateria” from a coworker that grew up in Lake Charles, Louisiana and went to school at Texas A&M. I assumed it was a localism.
I’ve heard it used, but assumed it was a slang expression. I didn’t know there were people who used it unironically to mean “laundromat”.
The alternate (and non-Westinghouse) spelling ‘laundrymat’ was in occasional use at least as early as 1903.
Why yes. I’ve heard it. 
I may have invented it 
The place I used in the 80s in San Diego was open for 24 hours and was attached to a 7-11. It was a party.
I’ve encountered the term “Washateria”, but I couldn’t tell you where. Not in any of the places the OP points out.
This thread interested me because it never occurred to me that washateria was a regional word, so I have started a thread looking for similar regionalism.
It occurs to me as I type this this very second, that I could have just sent this thread off on that tangent, but here we are…
I probably found it in a book somewhere.
At some point (1950s) the term ‘washerette’ also started being used for laundromats. However. . .
There was indeed a portable washing machine in the 1920s called “The Washerette.” Have not been able to identify who manufactured it, but the ads for this breadbox-sized doohickey tout:
The Washerette weighs but six pounds and may therefore be easily carried about. In fact, it is so small you can put it in your trunk. Just attach the hose to any faucet and the water will lubricate and run it. The result: Your finest garments will be washed spotlessly clean without the slightest wear on your clothes or effort on your part.
I also saw an add for a car dealership offering jobs to “washerettes” and “greaserettes,” who would respectively wash and lube vehicles. Those didn’t seem to catch on anywhere else, though.
I heard it when I was living in Lafayette, Louisiana. That fits with the southern idea.
Best combo I saw was a combination laundromat/video game arcade near the University of Pennsylvania campus in Philadelphia. That was like 40 years ago, don’t know if the arcade is still there now.