If we were approved in writing to get out of our lease early, but we did not sign the buy out agreement (with the dollar amount in it to get out of the lease), are we bound by the buy out agreement?

It depends - did you move in anyway? Or did this happen after you had a lease and were living there? A timeline showing the context is needed.

We lived in the apartment for 8 months of the 12 month lease. We signed a Notice to Vacate in June for 7/31and were approved but were never given the buy out agreement to sign. Now they are saying we owe 2 months rent which is what the buy out agreement says

It may have been foolish of them to approve the vacate notice, but no judge is going to let you get away with saying you don't owe anything because you didn't sign the buy out. The notice to vacate was contingent on agreeing to a buy out. Technicalities like this only work in criminal cases; rarely in a civil case, and probably never in housing cases.