Hey all, I realized I should update this thread with things I've found out in the last 9 months, in case anyone else has the same issue.
To answer my own question, the answer is to start searching on Medicare.gov. You can do a map search for nursing homes and set filters for "Accepts Medicaid". Government ratings from inspections are built right in, too! From there you make a list and start calling.
If you live in a HCOL area, prepare to search elsewhere. Of all my local places, not a single facility had an open Medicaid bed. 1/3 had waiting lists. (In my area they were all over a year long.) 1/3 didn't have waiting lists and you just had to call at the right time. And 1/3 absolutely never called me back no matter how many messages I left, and the only way to get an answer was to go into their offices and find the admissions person face-to-face, who usually told me they were a "no waiting list" facility. I'm now looking at places over an hour drive away from me. This is because...
No one actually wants Medicaid patients. They don't make as much money on them as private pay patients. The only reason they have Medicaid beds at all is because the state makes them. But they certainly don't want to recruit Medicaid patients, and if they DO have X number of beds set aside for Medicaid they will, if at all possible, hold them for their own private pay residents who run out of money and go onto Medicaid. On a related note...
Senior Care Consultants aren't much help. These people exist to help private pay patients and they get a cut from the facilities they refer them to. You don't pay them, Sunset Villas does for bringing them a customer. If you're looking on behalf of a Medicaid patient you have to pay the consultant out of pocket, which costs thousands of dollars.
So, right now I'm applying to a place over an hour from me, which still has a waiting list, but a shorter one than any facility near me. I have two backup options, neither of which I'm crazy about, but the alternative is leaving my dad in Illinois and flying out there on a regular basis, because he needs a competent adult keeping tabs on him and making sure he's being taken care of. Every option kinda sucks.