This is very simple. You worried too much

You give her the lease to (draft) show good will.
Of course you don't sign it. Landlord never signs until the tenant signs.

When she returns her signed version, you hold off sending back the executed one (with your signature) until the check is completely clear.

Sometimes tenant acts slowly. It may even take 3-5 days for her to get back to you.

Alternative is not to accept personal check, ask her to send cashier's check