Air Carrier Captain reported altitude deviations due to the loss of VNAV Path and a malfunction with the Autopilot with regard to directional control. After resetting the Autopilot the control problem was resolved and the approach completed.

Narrative

While beginning the ZZZZZ 2 Arrival into ZZZ1 we were given instructions to expedite our descent to FL200. I decided to use FLCH because the airplane wasn't descending quickly enough in VNAV. We were given another step down altitude of 17;000 feet prior to ZZZZZ1 and told to fly the SID as published at this point. After leveling at 17;000 the automation was out of VNAV at this point because I used FLCH for the expedited descents. I was late in recognizing that the aircraft wasn't descending on path and as a result we missed our crossing altitude at ZZZZZ1 by around 800 ft. I believe we crossed ZZZZZ1 at or near 14;800 ft.I don't normally use FLCH while flying STARs; but in this case I elected to do so with the expedited descent. Looking back I should have just deleted the crossing restrictions leading up to ZZZZZ1 because the airplane was descending on VNAV Path based on these and was descending too slowly. After leveling at 17;000 ft. as per ATC instructions we put 10;000 ft. into the Mode Control Panel (MCP) to fly the SID as published. I was late in recognizing we were not in VNAV and not following the descent path.Approaching ZZZZZ2 at 11;200 ft. it became apparent that the airplane had no intention of leveling off to meet the altitude restriction of 11;000 ft. I quickly disengaged the Autopilot and began to level off. I was surprised that the airplane took so long to stop the descent. We leveled off at 10;600 ft. While turning left to follow the flight directors I immediately noticed the control yoke wouldn't move beyond a certain point. Control travel was limited to perhaps a quarter of normal travel. Same with trying to turn to the right. Any attempt to move the yolk past this point was met with strong resistance. I tried a more aggressive turn to the left and the best I could get was perhaps a 3 degree turn. At this point I had my FO try his controls. His were the same. We briefly discussed the situation and I decided to turn the Autopilot back on to see how it performed. We planned at this point to continue in and see how the airplane handled at slower airspeeds and [request priority handling] if it was the same. After slowing down and deploying flaps I turned off the Autopilot. I should mention the Autopilot didn't seem to have a problem with the few turns we had; just with following the VNAV Path. After turning off the Autopilot the flight controls acted normally and we were able to fly the ILS XXR approach and land without any problems. We didn't have to [request priority handling]. I'm not sure what caused the flight control issue. It felt as if there were bungie cords attached to the yolk and I was pulling against them. Kind of an elastic strong resistance with a little give. There might have been a pitch problem to a lesser extent. The airplane had a tendency to remain high on path and completely failed to level off prior to ZZZZZ2. It also took longer to stop the descent than it normally would.