Provide solid information to the new landlord and their agent. Show you gave proper notice to vacate, your rents checks were on time (copies of cleared checks) and maybe photos of your current place to show you keep it clean. Remember, the new landlord will check your last 2 rental homes. The one before hopefully will give you some good references.
Your lease would allow showings during your last 30 days typically. Make sure your allowing those showings and not getting in the way. Other than that, I think you might have a case against the landlord. Credit companies and reporting firms like Trans Union cannot put out false statements about you. I think a landlord would be under the same conditions. Good Luck!