There's not all that much you can do to protect the condition of the inside the of home. There may be damages and that goes along with inherited tenants and evictions. You hope that when you bring new tenants in you avoid or reduce those chances going forward through your screening process.
Do the terms of the lease automatically renew or have the tenants give notice to vacate or have you sent a termination of tenancy? It sounds like you expect the tenants to pay rent and vacate at the end of the month, but that may not necessarily be the case.
As @Anthony Thompson said, the safest day to send your 5 day demand is on the 17th of the month. I have been sending mine out on the 16th and haven't had any issues in district court in Warwick.
I've worked with Sean O'Leary with O'Leary Murphy Law out of East Greenwich for years. He focuses on evictions, among other things. For all my evictions, I've gotten possession and a money judgment, not one or the other?
Don't refund the security deposit to your current tenants early. The point of the security deposit is to cover any damages to the unit or cover any outstanding balances at move out. You are required to refund the security deposit within 20 days of move out provided that you have been given a forwarding address to send it to.