DURING PREPARATION FOR AN ILS CAT II AUTOLAND APCH TO DFW; SITUATIONAL AWARENESS WAS LOST. XFER OF CTL BTWN PLTS AND XFER OF CTL FROM ONE AUTOPLT TO THE OTHER RESULTED IN NO ONE FLYING THE ACFT AND DEV FROM TRACK COUPLED WITH AN ALT OVERSHOOT.

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ON ARR INTO DFW; WX WAS 1/4 MI VISIBILITY BROKEN; 100 OVCST; RVR 3000-4000 FT AND DECREASING; SO THE CAPT DECIDED TO BRIEF FOR A CAT II AUTOLAND APCH. WHEN THE CAPT (PF; LNDG) XFERRED THE ACFT TO ME (PNF; NOT LNDG) I PLANNED TO SWITCH THE AUTOPLT FROM #1 TO #2. AT THE SAME TIME THE CAPT XFERRED CTL OF THE ACFT TO ME; WE WERE GIVEN INSTRUCTIONS TO DSND FROM 6000 FT TO 5000 FT; SLOW TO 210 KTS AND MAINTAIN HDG (AS OPPOSED TO STAYING ON THE ARR). WHILE COMPLYING WITH THE NEW INSTRUCTIONS; I THOUGHT I HAD RE-ENGAGED THE AUTOPLT. HOWEVER; THE AUTOPLT DID NOT ENGAGE; AND WHILE I LOOKED DOWN AT MY APCH PLATE THE ACFT ENTERED A SHALLOW L TURN AND DSNDED BELOW 5000 FT. AFTER A MOMENT OF CONFUSION (THINKING THAT THE AUTOPLT WAS ON) I IMMEDIATELY BEGAN CORRECTING TO THE ASSIGNED HDG/ALT. A FEW MOMENTS LATER; ATC GAVE US A ROUTINE TURN AND SAID NOTHING OF THE DEV. LESSON LEARNED: JUST LIKE WHEN ACFT CTL IS PASSED BTWN PLTS; WE REQUIRE A POSITIVE RESPONSE. FROM NOW ON; I WILL MAKE SURE THAT AUTOPLT IS PROPERLY ENGAGED AND FLYING THE JET. SUPPLEMENTAL INFO FROM ACN 495488: ACFT BEGAN AN UNCOMMANDED L TURN. CAPT REDIRECTED THE FO R TURN TO 230 DEGS. FO REPEATEDLY PULLED HDG SELECT WITHOUT RESPONSE AND DSNDED THROUGH 4000 FT. CAPT AGGRESSIVELY TOLD FO TO GET BACK ON ALT. 3 TO 4 TIMES CAPT REDIRECTED FO TO CORRECT BACK TO 4000 FT. DSCNT STOPPED AT 3400 FT. FO WAS BUSY WITH OTHER COCKPIT DUTIES WHEN ACFT BEGAN L TURN. WHEN CAPT BROUGHT HIS ATTN BACK TO ACFT CTL AND ACFT WOULD NOT RESPOND TO HDG SELECT; ALTDEV OCCURRED. NEW FO WAS OUT OF SITUATIONAL AWARENESS LOOP.