This potential buyer has advised that he would buy my investment property in Alabama so that his college age son could live in one of the units if it was vacant, but both sides are rented. The agreement with the management company expires upon sale of the property. Could I offer one of the current tenants their full security deposit plus a few hundred dollars to move out?

The property management agreement is irelivent. What does the lease with the tenants say? If it says they can not be kicked out then they get to stay.

You can offer the tenants anything you want and if you come to a deal with them to get out early make sure you get it in writing.