*IF* this is the case in your state, you might as well accept any partial payment they offer. You cannot file to evict this month. Next month refuse a partial payment, serve the Pay or Quit and then file to evict.

If your state allow you to accept partial payments and still evict, it is up to you if you want to accept the partial payment. (But why not if you can still evict?)

I see nothing in the statute that says that you have to accept rent up to the date of court.

I also don't understand why you'd give these tenants 3 months of notice to move out? They don't sound like ideal tenants and that will give them a long time to trash the place. Or they will mistake your notice for a vacate notice and move out immediately, leaving you with several month of rent owed for the end of the lease. Why not wait until one day after the lease expires, then serve a 7 day notice to vacate on the month to month tenant?