TCASII ALTDEV COMPLYING WITH TCASII COMMAND.

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DURING DEP FROM LGA ARPT TWR INSTRUCTED US TO; 'TURN TO HDG 350 DEG NOW.' WE INITIATED TURN R OVER RWY 22 - RWY 31 INTXN. UPON CLBING OUT OF 15000 FT MSL (AFTER CONTACTING DEP AND CLBING TO 12000 FT) WE RECEIVED AN RA; DSND COMMAND FROM TCASII. I INITIATED DSCNT IMMEDIATELY AND LOOKED AT TCASII TO SEE 2 TARGETS. THE FIRST TARGET WAS AT 2:00 O'CLOCK AT 2.5 MI AND 300 FT ABOVE OUR ALT. THE SECOND WAS AT 8-9 O'CLOCK AND AT 3 MI AT OUR ALT. THE CAPT (PNF) TOLD DEP WE RECEIVED AN RA AND WE DSNDED TO COMPLY. DEP INFORMED US WE HAD 1 ACFT NOW BEHIND US AND 1 AT 8-9 O'CLOCK AT 1000 FT COMING UP ON THE HUDSON; NEITHER A FACTOR; CLB TO 12000 FT. THE CAPT NOR I VISUALLY CONTACTED ANY TFC; SO I INITIATED CLB TO 12000 FT. OUT OF 1800 FT MSL; I RECEIVED ANOTHER RA TO DSND. I WAS HESITANT. UPON RECEIVING 'DSND NOW' COMMAND. I DSNDED AGAIN TO APPROX 1200 FT MSL. WE WERE BLOW THE CLOUDS AND PERCEIVED NO IMMEDIATE TERRAIN OR ACFT COLLISION DANGER. AFTER CONTACTING DEP AGAIN; HE TOLD US 'NO TFC NOW; YOUR TCASII; SOMETHING MUST BE WRONG WITH IT. MAYBE YOU SHOULD TURN IT OFF.' CAPT AND I CONCURRED AND IMMEDIATELY INITIATED AGGRESSIVE CLB TO 12000 FT AND CYCLED XPONDER AND TCASII. AFTER LEVEL OFF; I TRIED TCASII TEST AND IT TESTED OK. CONFLICT MAY HAVE BEEN WITH ACFT ON VISIBILITY OR LDA-A TO RWY 22. MAYBE NEW YORK SHOULD NOT ISSUE TURN TO N UNTIL YOU'RE OUT OF CERTAIN ALT AND AT END OF RWY 31. ALSO; BEFORE TKOF; RADAR STABILIZER WENT INOP AND SCREEN SAID FLT 6. QUESTION MAY BE WE HAD A SHORT THAT CAUSED STABILIZER INOP AND TCASII TO MALFUNCTION AFTER FIRST RA. I BELIEVE FIRST RA WAS CORRECT. WITH INFO WE HAD; WE DID OUR BEST.