SA-227 Captain reported rising oil temperatures required the in-flight shutdown of one of the engines. Due to deteriorating weather; the crew elected to continue to the destination airport with assistance form ATC.

Narrative

During cruise I noticed the oil temperature for the right hand engine was starting to climb into the top of the green and was abnormal so I decided to keep an eye on it. The oil temperature continued to climb and once it entered the yellow; I ran the QRH and reduced engine power. With reduced engine power it continued to climb so I brought power all the way back to flight idle. With power at flight idle the oil temperature went above red line and I performed a precautionary shutdown. After shutting down I advised Center. They asked if I would like to land somewhere closer as opposed to my destination but the weather was really bad in ZZZ and ZZZ1 was looking like it would still be the best option due to weather. My airplane was also maintaining a decent speed and altitude. My best option was to continue on to ZZZ1.I shot an approach into ZZZ1 and rolled off the runway onto a taxiway before coming to a stop and shutting down so I could clear their runway and not close it down. I then got towed over to the MX hangar by the FBO. I detected the event when my scan revealed an abnormal oil temperature indication. Some malfunctioning part within the right engine causing high oil temperature. I followed company procedures and ran the QRH; performed a precautionary shutdown; requested priority handling; and landed uneventfully. Unsure what caused it in the first place; this will be something for MX to figure out.