Trying To Stay In Apartment For Full 60 Day Notice Period

I live in MA and recently signed on a new apartment. Per our current lease, we're required to give 60 days written notice if we plan to move out. If the last day of this notice falls after the last day of the lease, we're responsible for paying through the notice at the month to month rate.

Our current lease ends on 6/1, but we forgot to provide written notice to our landlord until 4/11. When we did provide notice, we said that we would move out at the end of the lease, hoping that they wouldn't enforce the auto-renewal and 60-day notice period.

Now, we're now being charged for an additional 7 days at the new rate + $1000 (prorated). Since we're paying for that time, we were hoping to use the additional days to extend our move-out. However, the leasing office said that since we provided a move-out date of 6/1, they have a tenant moving in on 6/8, and the apartment needs 5-7 days to turn over the unit.

Is there anything we can do to get out of paying for the days that we're required to be out of the unit for turnover, or to stay longer than our original move-out date?