P180 Avanti pilot reported observing Decreasing Left Engine Oil Pressure and Stable Oil Temperature on a post maintenance check flight. The pilot requested vectors for a return to the field. Engine Oil Pressure continued to Decrease. The pilot performed an in flight shut down of the Left Engine and requested priority handling. A precautionary landing was made; with directional control not maintained; resulting in a runway excursion. The pilot secured the aircraft and waited for assistance.

Narrative

After takeoff and climbing through FL270 to FL320 on an westerly heading I noticed the oil pressure on the left engine decreasing slowly. I asked ATC to level off at FL280 to investigate. Upon level off at 280 the oil pressure was continuing to slowly decrease; oil temperature was stable; suggesting a possible sensor problem. I asked ATC for a return to ZZZ. I was advised to turn to a heading of 140 degrees and descend to FL240. Oil pressure continued to decrease; oil temp. still stable. I reduced power on engine as per the checklist. Oil pressure still continued to decrease and temp stable. On this heading for approx. 2 minutes I felt a slight 'bump' through airplane. Thinking an engine failure; I reduced power to idle and feathered the prop. I [requested priority handling] and requested direct to ZZZ and a descent. Cleared to 10;000 ft. and direct to ZZZ. After further analysis of engine instruments engine had not failed. I was given vectors to final on Runway XXL and cleared to land. I landed safely but just after landing on rollout had a controllability issue. I safely got the airplane under control and then applied MAX braking. As a result I was unable to slow the airplane to a safe taxi speed to exit the Runway by the last Taxiway. I subsequently ran off the movement area into the overrun. No damage was sustained by the airplane or airport property. The action of [requesting priority handling] was a little premature at the time and I should have declared a precautionary instead. I did everything I could to slow airplane down to a safe taxi speed. Trying to maneuver at a speed faster than a safe taxi speed might have produced some unwanted problems.