A landlord deducting for damages should be clever enough to include a "payment in full" on such a check. If not, he's not a very bright landlord, and should not be in the landlording business.
In the past, it was recommended that someone cashing such a check marked "payment in full" should cross out the "payment in full", and include "under protest", with "accepted as partial" when endorsing it, though recently, I understand even this method may be questionable.
When I took a seminar on this issue once, I was told that state laws and precedents apply, so there is really no real universal rule. The safest universal rule suggested in not to cash it. I was in the "credit and collection" field in one of my previous careers, and we often receive partial payments from customers marked "payment in full" on them.
To be on the safe side, if the OP writes anything on the check, make a copy of the check before cashing it, in the event the maker makes any additional changes on the check before going to court, assuming he gets the checks back. I saw a court case once where the maker wrote a few more things on the check after getting it back to bolster his case.

