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.