A Center Controller reported an Air Carrier which they were not communicating with descended from its assigned altitude. The aircraft which had shut down one engine had acknowledged a frequency change via CPDLC but remained on the previous sector's frequency.

Narrative

Aircraft X entered airspace from a ZZZ1 sector XX. Aircraft showed CPDLC eligibility. I thought the aircraft checked on and I toggled the aircraft on frequency since I showed eligibility after the arrow indicating the aircraft took the frequency change. The aircraft was still on the sector XX frequency. West of ZZZ the aircraft [requested priority handling] on their sector frequency and the pilot started a descent. This is where I was calling the Aircraft X with no response. After two calls and no response I went into action moving other aircraft out of the way and verbally told sector XY to start moving aircraft as they had started flashing with traffic since the descent was not in the data block and the computer then obviously flashes with any possible traffic.Sector XX called on the line and said Aircraft X [requested priority handling] and was in a descent and I told them to tell the pilot to maintain 35000 ft. They replied the aircraft was on the way over. When the pilot checked on I told them to turn 20 left and state nature of [the problem]. They replied they were in the process of shutting down an engine. I told them to maintain 35000 ft. And they wilco'd. The pilot stated they might land ZZZ. Supervisor was notified during this and promptly put in a D side. No separation was lost.I realized after the fact the pilot can acknowledge a frequency change and CPDLC will show that they took the frequency change and that I have eligibility but that does not mean the pilot actually tuned to that frequency. This is something that could be clarified or refreshed to other controllers.