CAPT OF ACR MLG ACFT DEVIATED FROM ASSIGNED ALT IN RESPONSE TO A TCASII RA.

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WE WERE AT FL280; CAPT FLYING. AN OPPOSITE DIRECTION ACFT WAS CLBING TO FL270; AND WAS CALLED TO US AS TFC BY CTR. WE RECEIVED A 'TFC' WARNING ON THE TCASII. I SPOTTED TFC; WHICH APPEARED WOULD PASS SAFELY TO OUR L; CAPT DID NOT SEE IT. WE RECEIVED A 'CLB' RA FROM THE TCASII WHICH WE FOLLOWED USING THE DIGITAL FLT GUIDANCE SYS. AT 300 FT ABOVE ASSIGNED ALT; A 'MONITOR VERT SPD' RA WAS RECEIVED; AND WE STOPPED THE CLB TO RETURN TO ASSIGNED ALT. BECAUSE WE WERE 'HVY' AT THE TIME; WE CRUISE PWR SETTING MADE IT DIFFICULT; AFTER THE RA; WITH LOSS OF AIRSPD IN THE CLB; TO RECOVER RIGHT ON ASSIGNED ALT; WENT BELOW ABOUT 250 FT; THEN BACK UP TO 280. BTWN THE 'CLB' RA AND 'ALT' AURAL WARNINGS IT WAS DIFFICULT TO COM WITH ATC ABOUT WHAT WE WERE DOING. I HAD TO EXPLAIN 3 TIMES. IT APPEARS THAT THE HVT ACFT OVERSHOT HIS ALT BY 300 FT. THIS IS WHAT TRIGGERED THE RA IN THE FIRST PLACE. AT THE TIME THE CLB RA WAS ISSUED THE TCASII SCREEN PLACE. AT THE TIME THE CLB RA WAS ISSUED THE TCASII SCREEN SHOWED TFC 700 FT BELOW US WHEN WE WERE AT 280. I THINK THE CAPT'S RESPONSE WAS PROPER IN THAT THE TFC I SAW MAY NOT HAVE BEEN THE SAME AS WARNED OF BY THE TCASII. ALSO I'VE SEEN MANY ACFT PASS THAT WERE NEVER CALLED BY ATC.