So in my limited experience, you just had to submit continuing eligibility reports that were due every June IIRC. I was actually on Kaiser myself, and your provider doesn’t change unless you change it.
I was on it for financial reasons for a while pre-COVID, so my memory is fuzzy and it may have changed. I could look through the DHCS website to confirm, but hopefully this helps.
EDIT: Here’s the official website, which isn’t necessarily detailed yet seems to confirm what I said. They generally try to make it easy to stay on Medi-Cal if you need it. Also, here’s a page linking to a bunch of general Medi-Cal info. I know that it can be daunting navigating the government websites, but they’re generally the best places to find information.
I think Eternity requires an instance to be running v19 of Lemmy, and Beehaw stopped at 18.x, so that’s why it doesn’t work.