Is it possible to get a case manager for my mother who is on Medicare? What kinds of things do they do?

Goose7: Pose your question to the head of the AL.

So sorry you are going through stressful caregiving. Just want to clarify in my mind... your Mom is in a short term rehabilitation facility for skilled nursing or rehab or is she in an Assisted Living facility? Does she have traditional Medicare or a Medicare Advantage program?
I'm curious because Medicare, as a health care insurance, doesn't pay for a stay in Assisted living (unless it is an approved 10 day respite care situation). Who signed her into the current facility? Does she have a dx of dementia? If she is in a short term care facility for skilled nursing or rehab and was sent there by her physician after a 3 night stay admission in a hospital, Medicare will pay for her stay for the first 20 day provided she is participating and improving. If the staff determines that she requires additional skilled nursing/rehab, traditional Medicare will pay 80% of the costs until day 100. If she has Advantage Medicare the lengths of stay will depend on the individual policy.
You can always hire a geriatric case manager but, in general, they only evaluate the patient and the patient's living quarters. They may make recommendations for supplies but don't go to get them. Sounds like you may be thinking of a caregiver who might (depending on the contract) pick up supplies, fix food, do laundry etc. Depending on your state, a caregiver who administers meds, versus someone who simply reminds her to take her meds, may need a different license and so will be a bit more expensive.
f she is in a STR (short term rehab) facility, they have a responsibility to discharge to a safe environment. She has a house to go to but may not be safe. You can meet with the administrator and voice your concerns and tell them that you are moving shortly (not sure if you mentioned if you have PoA). Unless she has quite a bit of money, she will need to apply for Medicaid for a stay in a long term care facility. Medicaid will do a 3-5 year look over the shoulder to make sure she has not gifted money to anyone in order to qualify; at some point, the house will need to be sold so that Medicaid can recoup the funds they have spent on her care in the facility. If you haven't keep good records of her expenses in the past - start NOW!! Once she is in an LTC, facility they will get all the supplies she needs and will order her meds and actually administer them because they have certified nursing staff 24/7. Medicaid generally will pay only for shared accommodations, so she will have a "roomie" whether she likes it or not.
You are right to consider getting a job because you need to start saving for your own "old age" because social security alone will not suffice and Medicare, while good, is not free.
Good luck on this part of your journey.

Are you her PoA or legal guardian? If so, is the PoA authority in effect (ie the criteria has been met, such as a medical diagnosis of impairment?)

If you aren’t her PoA and she is has enough cognitive decline you should have this discussion with the facility admins and/or social services for that county. If you remove yourself completely this allows social workers to come in to assess her situation. Eventually she can be assigned a court-appointed guardian who will manage all her affairs and you can just have whatever relationship with her that you want, minus the oversight and responsibilities.

However, you won’t be allowed to live in her house anymore. So, you have a solution…you just have to be willing to accept it.

I had nothing to do with Moms prescriptions. They were all monitored by the RN. Mom had to use theirvpharmacy. When refills were needed, she spoke to Moms PCP. The residents were not allowed medications in their rooms.

Are you talking about Medicare or Medicaid, because the former doesn't have case managers while the latter "may" in some instances.
I would go to the administration of your mother's care facility to discuss all of this with her.
As to returning her to her home I think that is not a good idea.
I encourage you to now begin to remove as much of this from your shoulders as you are able to.
You may still live in your mother's home, and you say your children are grown, but you need now to know that if Mom is on Medicaid then there may be clawback upon her death to recover assets she currently has in her home.
You should see an elder law attorney to ascertain where you stand in all this, and then make decisions for your own future, one of which is to attempt to work as hard as you can as much as you can to save up a nestegg for the time you, yourself may be in need.

I sure wish you the best.