My lease expires July 1st and when I went in to pay my June rent I spoke to the head leasing agent and asked her what we needed to do come July. She said once we vacated to return the keys and they had 21 days to refund our deposit. Now, they are telling me they needed a 30 day notice in writing and we are responsible for July rent. Is this right?

Reread your lease carefully, the need for a 30 day wtitten notice has to be in the lease )quite common) if they intend to hold you to it. If it's there you are stuck for another month as a month to month tenant.