B737 Captain reported pressurization failure and diversion to a safe landing.

Narrative

Arriving at the aircraft; we noticed the MEL for the pressurization system. On the MEL; the actions for the flight crew were to leave the controller in the ALTN mode for all phases of flight because of the system malfunctioning in AUTO mode causing the AUTO FAIL light to illuminate. After departure from ZZZ; climbing out of 6;500 ft we had an AUTO FAIL light come on with the pressurization controller in ALTN mode. Following the MEL procedure requires the crew to switch from ALTN to MAN mode for the pressurization controller.After complying with the MEL procedure; the pressurization system started to over pressurize and the manual mode was not working correctly. We advised ATC to level at 9;000 ft so we could work out the pressurization issue without climbing above 10;000 ft. After reading through and attempting to accomplish the FM (Flight Manual) checklist for the AUTO FAIL light; the controller in manual mode was uncontrollable showing erratic fluctuations in cabin altitude and the differential pressure was maxed out over 9 PSI creating extreme discomfort in head and ears. We as a crew then made the decision to divert to ZZZ1 with coordination with dispatch and ATC. After the decision was made to divert; we decided it was best to request priority handling. Once we descended to pattern altitude for ZZZ1 (2000ft); the checklist requires to open the outflow valve to the full open position for landing. This action worked as it should and was able to land the aircraft safely and unpressurized. In manual mode it seemed the outflow valve was either open or closed. We used slow descents less than 1000 fpm and we slowed for drag extension to burn fuel for under landing weight.