DURING AN INCORRECT DSCNT AND CLB BACK; A B737-300 CREW RECEIVE A TCASII FROM A XING ACFT.

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WE WERE LEVEL AT FL280 APCHING YOUNGSTOWN ON J211. THE CAPT CLRED OFF RADIO #1 TO GET ATIS. WHILE HE WAS OFF; I RECEIVED A CLRNC TO DSND TO FL220. I ACKNOWLEDGED THE DSCNT TO FL220 AND SET IT IN THE MCP. I THEN BEGAN THE DSCNT TO FL220. WHEN THE CAPT RETURNED I INFORMED HIM WE WERE CLRED TO FL220. AFTER LEVELING AT FL220 ZOB ASKED US WHAT ALT WE WERE AT AND WE RESPONDED FL220. HE THEN SAID WE WERE CLRED TO FL260 AND TO CLB TO FL260. WE BEGAN A CLB TO FL260. PASSING THROUGH FL240 CTR THEN DIRECTED US TO RETURN TO FL220 WHICH WE DID. CTR THEN POINTED OUT TFC AT 1 O'CLOCK POS; PASSING FROM L TO R. THE TCASII ALSO GAVE US A TA. WE SAW THE TFC AND DID NOT HAVE TO TAKE ACTION TO AVOID IT. THE CAPT THEN NOTED THE DETAILS OF THE EVENT INCLUDING THE CTLR'S PHONE NUMBER AND INITIALS. THE REST OF THE FLT WAS UNEVENTFUL. SUPPLEMENTAL INFO FROM ACN 596820: I WAS OFF ATC FREQ TO GET THE CLE ATIS AND CAME BACK AND WAS ADVISED WE WERE CLRED TO FL220 BY THE FO (PF). WE WERE AT FL220 FOR SEVERAL MINS AND THE CTLR ASKED OUR ALT AND I ADVISED FL220. HE IMMEDIATELY TOLD US TO MAINTAIN FL260. I SAID OUR CLRNC WAS FL220 AND HE SAID MAINTAIN FL260. WE STARTED A CLB TO FL260 AND PASSING FL240 WE GOT A TA AND ABOUT THE SAME TIME THE CTLR TOLD US TO MAINTAIN FL220. WE STARTED DOWN IMMEDIATELY TO AVOID THE RA AND CONTINUED ON AT FL220. I ASKED MY FO IF THE CLRNC WAS FL220 AND DID HE READ BACK FL220; AND HE SAID THAT HE HAD HEARD AND READ BACK FL220. WE CONTINUED ON TO CLE WITHOUT EVENT. UPON REACHING CLE I CALLED COMPANY ATC SPECIALISTS AND HAD HIM CONTACT ATC. WE NEED ACARS ON ALL ACFT.