Problem tenants gave 30 day notice! How hard should I push to get last month's rent?

[Johann Jells]

My inherited problem tenants (loud, bathroom floods, calling city authorities on me) have given notice. I'm thrilled they're leaving, I was considering offering them a few grand to go.

But they have not paid June rent. My wife the social worker says don't push it, it might make the difference of them actually getting a new place. I've read here some landlords will pay to get a place back in decent condition from problem tenants on moveout. Is she right? With 4 kids I'll bet there's a decent amount of damage.

When you file an eviction and the tenant moves out before action, does the eviction still go on their record or does it become a small claims issue?