SAV Tower Controller reported clearing an aircraft for takeoff before another aircraft had cleared the runway. The reporter stated fatigue related to frequent overtime as a result of short-staffing was a factor.

Narrative

Aircraft Y cleared to land Runway 10; Aircraft X holding short of Runway 10 with a CFR (Call for Release) time. A third aircraft was initiating an intersection departure from Runway 19. Ground had multiple crossing requests for Runway 19. Approach on hotline to coordinate something but was indistinguishable from helicopter calling with request on frequency.I gave a takeoff clearance to Aircraft X to meet his flow time but only completed a partial scan due to other operational activities. Aircraft Y must have been on short final; dropped from the radar display. I became aware of my mistake only when Aircraft Y requested exit instructions for his FBO. I located the aircraft; issued the exit instructions; and observed that Aircraft X was still a few seconds away from beginning takeoff roll. At that point I could see that Aircraft Y would exit before separation would be lost. Aircraft X questioned the takeoff clearance; and I reissued that.I believe mental mistake is attributable to fatigue caused by months of overtime assignments; and lack of staffing. No oversight in the Tower at the time; no one in the Tower cab noticed that I issued an unsafe clearance. Supervisor combined with Clearance Delivery and facing away from the operation.