As previously mentioned, the best course of actio is to refer this to an attorney.....but if the lease has expired, it has been reported to you by neighbors, that they moved out, and it appears there is no one there.....I'm uncertain why you would need to implement the eviction process?
Your contract says you can enter the property to show it and you can also enter it to deal with an emergency. If your decision it to enter the home, we would suggest you do so with a credible witness...a policeman, fireman, plumber, etc.
Again, your best bet is to speak with an attorney or the police before doing anything.
Deryk is right, don't do anything without bouncing it off a lawyer that knows this side of the business - it's amazing how many "rights" tenants have even when they pull this garbage. Good luck TJ....
I would also recommend contacting a good tenant eviction attorney. You probably still have to go through either an eviction process or at the very least a "notice of abandonment" process before taking possession. You should have the right to access the property for inspection to at least verify the current condition of the property but if I were you I would seek competent legal advice before attempting to take possession of the property.
We own several rental properties and manage properties for other investors as well. A lot of your questions have to do with the exact wording of the lease you had with your tenants...the rest of the answers can probably be found in the Georgia tenant/landlord code. Here are some helpful links: