BNA TRACON Controller reported having two aircraft on converging headings at the same altitude. Controller stated they were working combined sectors due to insufficient staffing and were fatigued from working overtime every week.

Narrative

Working Departure Radar East and Arrival Radar East combined. Switched Aircraft X on long side arrival to final. Around that time Aircraft Y departed. When I RADAR Identified him; I thought I climbed and turned him. Aircraft did not commence turn and was on converging heading with Aircraft X on the arrival. When I noticed the failure to turn; I turned Aircraft Y away from the arrival. Aircraft Y responded immediately and turned away. In hindsight; I am uncertain if I issued the first turn to the departure or not. I am unsure if there was a loss or not. I recommend more staffing at the facility to open more positions; reduce fatigue; and distribute workload. 6 day work weeks for consecutive years take their toll on the human body and fatigue can set in without the individuals even recognizing it. Current SIDs/STARs at BNA are outdated and inefficient. These procedures create conflict between departures and arrivals. New procedures have been designed and should be implemented immediately.