B737-300. ZAB CTLR NOTES THAT THE ACFT WAS NOT PROCEEDING TO THE CLRED VOR.

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DURING FLT TO OAK; THE CAPT AND I WERE FLYING AT FL350 WHEN CTR CALLED AND ASKED IF WE WERE STILL ON A HEADING OR CLRED DIRECT TO ILC. THE CAPT RESPONDED THAT WE WERE PROCEEDING DIRECT TO ILC. I IMMEDIATELY CHKED OUR NAV TO CONFIRM THAT ILC WAS STILL SHOWN DIRECT IN THE FMS AND THAT LNAV WAS SELECTED. CTR INFORMED US THAT WE NEEDED TO TURN R 20 DEGS TO PROCEED DIRECT. AT THAT TIME I SELECTED ILC ONCE AGAIN TO PROCEED DIRECT. PRIOR TO THE SELECTION OF DIRECT ILC; OUR HSI SHOWED US DIRECT. UPON SELECTING DIRECT ONCE AGAIN; THERE WAS NO CHANGE IN HEADING AND NO CHANGE IN COURSE. I BEGAN A R HAND TURN USING HEADING SEL TO PROCEED IN THE DIRECTION THAT CTR HAD STATED; WHILE WE BEGAN TO LOOK AT OUR SIT. NEITHER AUTOPLT CORRECTED THE HEADING PROB AND WE CONCLUDED WE HAD AN IRS/FMS PROB. THE CAPT PULLED OUT THE QRH. THE QRH DID NOT LEAD TO A SOLUTION. ABOUT THIS TIME CTR CALLED US AND STATED THAT WE NEEDED TO COME AN ADDITIONAL 10 DEGS R. HE RESPONDED THAT WE WERE POSSIBLY HAVING A NAV PROB; WHILE I DIALED IN AN ADDITIONAL 10 DEGS R. THE CAPT REFERRED TO THE ACFT MANUAL WITHOUT SUCCESS. ONCE AGAIN CTR CALLED WITH AN ADDITIONAL HEADING CHANGE; WHICH I COMPLIED WITH. IT APPEARED THAT OUR HEADING SYS WAS EITHER COMPLETELY OFF OR DRIFTING SLOWLY. I PROCEEDED TO HAND FLY THE ACFT AS NEITHER AUTOPLT WOULD ENGAGE. I TOOK OVER THE RADIOS AND CONTINUED TO FLY AS THE CAPT CONTACTED THE DUTY PLT. WHILE THE CAPT WORKED THE PROB THROUGH COMPANY RADIO; I KEPT CTR INFORMED OF OUR SIT AND REQUESTED A BLOCK ALT OF FL350-FL370. COMPANY MAINT WAS NOT ABLE TO SUPPLY ANY ADDITIONAL INFO ON THE PROB; SO AT THIS TIME THE CAPT DECIDED THAT WE NEEDED TO STAY VMC AND THROUGH COORD WITH THE DISPATCHER; SLC WAS DETERMINED TO BE OUR BEST DIVERT. WX AT OAK WAS 1100 FT OVCST. THE CAPT THEN RESUMED CTL OF THE RADIOS AND REQUESTED THAT WE WANTED TO REMAIN VMC TO THE FIELD. ATC COMPLIED AND STATED THAT WE COULD DEVIATE AS NECESSARY TO STAY VMC. WE PROCEEDED TO SLC AND LANDED UNEVENTFULLY. MAINT MET THE CAPT AT THE GATE; AND AFTER A FEW MINS; WE WERE INFORMED THAT SLC HAD A SPARE ACFT FOR US TO CONTINUE TO OAK.