Air carrier cargo pilot reported an autopilot flight control malfunction message and degraded flight controllability issues at cruise and in the descent. The crew had difficulty maneuvering to intercept the localizer and executed a missed approach then returned for a safe landing.

Narrative

Prior to the ZZZZZ5 arrival inbound to ZZZ at FL360 we started the APU for 'green book' ETOPS check. Upon running the CENTER A/P disconnected with accompanying red Warning and EICAS message. Thinking that it was due to a buss ferry error we reengaged the A/P.During the decent via the ZZZZZ5 at about FL260 we received a discrete EICAS 'Autopilot'. Seeing that the aircraft did not appear to be following the FD guidance I disconnected the A/P. Upon doing so I inputted a turn opposite to the defending turn and thought that the A/P was still engaged. It seemed at first that my inputs were fighting a still engaged A/P. I asked the Captain to take control for his feedback and opinion. He also stated it was not normal and very hard to control. I took back control and turned the A/P off via the MCP bar. Still no change. It felt that I was suggesting a direction for the plane vs actually flying it. There were no other warnings; lights; messages; or indications suggesting there was an issue. Captain further inquired another crew member to see if he has seen this before. He did not.I suggested that we [request priority handling] due to the severely degraded controllability. It was suggested to try a different A/P but felt strongly to not try due to not knowing what was wrong and didn't know if doing so would further degrade maneuvering ability and we were presently able to control the aircraft.We continued the approach but had to go missed (by ATC) due to difficulty maneuvering for localizer intercept. We were vectored back around and made a successful approach and landing.Suggestions on how to correct the reported concern: ['1) To let ATC know we would like maybe half-standard turns.\n2) Inquiry to Boeing as to whether this has occurred before and if there is a corrective action that can be performed inflight.']