Advice needed; Mom’s money runs out in about 6 months. No other resources, she will have to enter a NH. Should we do application for NH now?

Everythings been sold or cashed in. She requires 24/7 care because of mobility but also has dementia. The private caregiving situation is getting harder to maintain, an agency is now providing 2 days of care. Her annual physical for Medicare is coming up. There is only 1 NH my brother & I want her in, it’s a very desirable place so it can be difficult to get in. It’s also close enough for us to visit very often. My mom knows this is coming & isn’t happy about it but there’s no choice. My thinking is we should get her paperwork submitted when she has her physical. The Dr. has no prob with this & I do have POA. A physical has to be given at the time the Dr. completes the NH paperwork. I’m afraid if we don’t submit paperwork now, Mom will REFUSE to go back to the Dr. for another physical in a few months or at least refuse the TB test. She's still aware enough to know what’s going on! It could end up the NH has a bed open immediately, or it could take months. I think it’s best to start the proceedings now, especially since I’m not in a high population area & the other NH close by has a terrible reputation (my grandmother was in it). If we end up with a NH 45 minutes away I can’t keep a good eye on Mom as I still have a homeschooled child at home. Am I right in proceeding now? I also have questions about applying for Medicaid & will post those separately. (Sorry so long!)