Does anyone know if nursing homes have a notary available to sign needed documents without taking resident outside of facility?

ASK them.

I know that there are mobile notary services, at least in my community. They do charge extra though.

They most always do not, but many have access to notaries who do this work as a volunteer, coming to the hospital or to nursing homes. You might want to contact notaries in the area yourself. I had this done when my brother was hospitalized. There was no charge. I made a donation. That was in So. California. I can't know if this service exists elsewhere, but the Social Worker in the facility should have the answer. Otherwise, just call notaries and pay their fee for coming; I do believe and hope you will find one.

I really don't know that specific question, however we have been working with an elder care lawyer remotely. They are able to have their own notaries witness document signings remotely over Zoom. It is an interesting process.