How to sign house sale contract?

This is a question you must ask an attorney. I would not count on what your realtor tells you. All states are different. This way, you are certain everything is done correctly and nothing will be questioned if and when Medicaid needs to be applied for.

So agreeing with Ahmijoy. You might be able to speak with an attorney on a consult basis (free) or Legal Aid. You need the correct answer.

There should be an attorney at the closing. Its usually one hired by the realtor. You can have your own and it maybe a good idea to protect your interests.

I don't think there is a tax liability if you buy another house within, I think, two years. If Mom is competent, I see no problem in both of you signing.
You are smart in the "gifting" part if you need Medicaid within five years. Like that the deed says "or" instead of "and". Those two words make a big difference in handling this kind of thing. I think consulting with a lawyer before closing is a good thing. Like said, don't take what a realtor says. They are not always correct.