LTSS OCCURRED WHILE A DEP CTLR WAS REISSUING ALT CLRNCS TO THIS RPTING B737 FLC. THE CREW RECEIVED A TCASII RA; BUT CONTINUED TO FOLLOW THE CTLR'S INSTRUCTIONS.

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ON CLBOUT FROM CLE; WHILE TALKING TO DEP CTL; WE WERE CLRED TO CLB TO 12000 FT. AT ABOUT 7300 FT; DEP TOLD US TO DSND TO 7000 FT (MORE PRECISELY; TO 'LEVEL OFF AT 7000 FT') IMMEDIATELY. I TOOK THE AUTOTHROTTLE OFF; PULLED THROTTLES TO IDLE AND PUSHED THE NOSE OVER. AT 7000 FT; WE NOTICED TFC ON THE TCASII AND THE DEP CTLR THEN TOLD US TO DSND IMMEDIATELY TO 6000 FT. AS WE CONTINUED DSCNT THROUGH 6900 FT; WE RECEIVED AN RA TELLING US TO 'CLB.' SINCE THE THROTTLES WERE ALREADY AT IDLE; WE WERE DSNDING AND DEP CTL WAS TELLING US TO 'IMMEDIATELY DSND;' WE CONTINUED THE DSCNT. TFC PASSED OFF OF OUR L 100 FT ABOVE AND 2 MI OFFSET. WE WERE RE-CLRED TO 12000 FT; DIRECT TO ACO; WHEN ANOTHER TA (ALSO AT 7000 FT) SOUNDED. DEP CTL ADVISED US TO 'TURN S IMMEDIATELY.' WE DID SO AND WERE CLRED OF TFC. FOLLOWING THE RA INSTRUCTIONS WOULD HAVE TAKEN TIME TO BRING UP THE THROTTLES; REVERSE THE DOWNWARD TREND OF THE ACFT; AND CLB BACK THROUGH THE THREAT ALT. IN MY OPINION; CONTINUING THE INITIAL CLB TO 12000 FT WOULD HAVE AVOIDED ALL PROBS.