B737-900 flight crew reported a stabilizer out of trim annunciation during cruise required a consult with company maintenance and continued onto the destination.

Narrative

During cruise; a steady Stab Out of Trim annunciator light illuminated. We accomplished the appropriate checklist. Contacted dispatch and the [flight operations] and determined that it was safe to continue the flight to ZZZ. Thanks to the [flight operations] inputs we also notified ZZZ1 center that we were non RVSM (due to no autopilot) and with their approval we were able to continue at FL340. On arrival into ZZZ we wrote up the malfunction. The avionics tech informed us that the aircraft had the same malfunction on DATE. The remedial action for that malfunction was to MEL both autopilots. Our Avionics tech was able to isolate that the fault was indeed in the A autopilot system and that the B would operate normally. After completing the appropriate documentation we continued our flight back to ZZZ2. Being able to consult with Dispatch and the [flight operations] was a force multiplier; it was the [flight operations] who remembered that our system failure would result in our no longer being able to be RVSM and our dispatcher was able to provide us with the information that if we had to descend from RVSM airspace we would still be able to complete the flight at the lower altitudes.I find it a bit concerning that there was a prior write up on DATE and they were unable to determine the system fault but instead just MEL'd both A/P's and sent it on its way. Our avionics tech was able to show us the message clearly showing that it was A/P A that was faulting and not B. The previous write up was at an outstation so there was probably time pressure involved to 'get the jet out' but if someone had taken a safety pause to run the appropriate check they would have isolated the fault instead of just MEL'ing the entire system. There was a moment of 'question' when operating the checklist. The first question statement 1) Is the stabilizer trimming? Yes; No. A point of discussion among us was that since we were in cruise and the air was calm we weren't sure if the stabilizer was or wasn't trimming? We eventually determined that by reducing airspeed there should be a change in stab trim. It was by doing this that we were able to determine that the trim was indeed not operating. I would recommend exploring the option of including a statement in step one to verify the trim operation before continuing to the next step. After working the problem from event to resolution on the ground and return to flight I am wondering as to the operation of the stab trim system and selected autopilots. Was this as it appears to be a case of A autopilot faulting but giving us the indication of a stab out of trim? If so then wouldn't changing to B autopilot have also cleared the fault such as the return to service MEL? Does there need to be a revisit to the Stabilizer out of trim checklist to address if it's an A/P fault as opposed to a Stab Trim?