To my knowledge all standard 737 A/P installations are at most CAT IIIa fail passive.
No, actually fail-operational A/Ps have been fitted to new 737s for quite a while. Here's a quote from the Rockwell Collins website:
The EDFCS-730 provides Dual Channel autoland in landing conditions down to CATIIIb and rudder controlled rollout. The EDFCS-730 has been the standard autopilot flight director system for all Boeing 737NG series of aircraft since January 2003.