Separation agreement not signed - can he refuse to sell home

We have been separated for 5 months. I am in what used to be our marital home and he moved to an apartment. We are both on the deed to the home but the mortgage is in only my name. I am in a position where I need to sell the home or I risk defaulting on the mortgage and possibly facing foreclosure.

He says that he is going to refuse to let me sell the home, despite what we agreed to in mediation. Our agreement is unsigned.

What laws protect me from having to keep this house and go into foreclosure?

On the flip side, his apartment lease is in both our names, with me as the primary. He has no recent job history and could not qualify for the apartment on his own, and it was the only way to get him out of the house. Can I get my name off the lease?

The agreement you reached in mediation is not enforceable. You need to file an action for Equitable Distribution, and you may schedule a hearing shortly after you file suit to ask the court to order the home sold.
As for the lease you should speak to the property management company to find out what options they can offer you.