Has anyone allowed Guardianship of their Mother to the state to allow them to help financially with Memory Care cost?

Unusual situation for the state to step in as guardian. Have you been in communication with mom and the place she is right now.

Do you have mom's POA's? Did a court award you guardianship? That is the only way you could become guardian. It is a court process.

Something is missing here.

I too don't understand. Who called to get the State involved, the hospital Social Worker? In my case I was Mom's POA and she received Medicaid to offset her care in LTC. This did not make the State guardian of my Mom. I still made all decisions concerning her care. The State just paid the bill.

If you give guardianship to the State, Mom is then a Ward of the State. You lose all control of where she goes and any money she has. You will have no say in her care. You may not be told where she is.

You better fully understand what is going on here. Guardianship for you usually means a court date in front of a Judge and its expensive.
If you allow the State to take over her care completely make sure that you will be given the guardians name and that you will be kept in the loop concerning Mom's care. That you will be able to visit. Get it in writing. If you are talking to anyone, get their email and confirm the conversation and what was said. If they don't answer back with corrections to the email, they have accepted it. I always put "Please confirm receipt of this email" Sometimes dealing with certain Government entities don't get a confirmation but I always ask.

The memory care will likely be funded through medicaid. This means that all of Mom's assets will go to the facility; I know you recognize that is as it should be. If she has other assets they will be eliminated, gathered, sold for her care. I don't understand what you say of guardian of person for your Mom's state, but in general definition as copied online is:
"A guardian of the person is one whose responsibilities are to care for their ward's physical and personal needs. ... In all personal decisions, it is expected that the guardian of the person will take into account their ward's wishes, as well as his or her physical needs and financial circumstances.Feb 12, 2020".
This will give you pretty much zero input on placement and other things, but will allow you information on health, perhaps decisions on health. You should get this very straight in WRITING just what this means.
I think in many states people would be a lot better off allowing state to take guardianship and responsibility for individuals they don't wish to do hands on caregiving for; it is almost impossible to control things from out of state, and to manage all of it.
Can you elaborate and tell us a bit about Mom's condition currently?

To my knowledge there can't be 2 guardians. It's either you or the state/county. If you are thinking about Medicaid paying for her MC, someone will need to fill out the app for her (and this may depend on the state's rules about its Medicaid program). In my state basically anyone can fill out the app (so it doesnt need to the the PoA or guardian) but this representative would need to have access to the necessary info (bank statements, medical bills, assets, properties, etc).

If you are your mother's Power of Attorney, then there is no need for the state to pursue guardianship since your mother already has a legal representative (you).

If you are neither her PoA nor her guardian, and she is incapacitated to manage her affairs or make decisions in her best interests, then "someone" will need to become her legal guardian and it can be the state. This would mean you lose all ability to make decisions on her behalf, or to have knowledge of her medical and financial affairs, and is permanent.

Does this answer your question?