ZMA controller described a TCAS conflict event reported by an air carrier on vectors; the reporter indicating he/she was distracted with flight plan processing and failed to note/issue traffic.

Narrative

A B712 departed RSW and I climbed the aircraft to FL230. I had an aircraft depart TPA Approach Airspace that was requesting a flight plan to MYAM. I was attempting to depart the flight plan for the aircraft but was unable to. Additionally; the aircraft was only requesting 9000 but he later advised me that he would be requesting FL230. I made the determination to keep the aircraft on my frequency and point out the aircraft to the low sector below me. I was not able to activate the flight plan because it was under the control of the sector below me which required me to /OK the flight plan. Next; Sector 07 called to effect coordination with me and about this time the B712 advised me that he was responding to a TCAS advisory. I turned the B712 to 270 and issued the traffic. The B712 questioned who was working the traffic and advised me that that was 'pretty close'. When I observed the traffic was clear of the B712 I cleared him back on course. At the time the B712 was issued FL230 I scanned along the route of flight and did not observe traffic. While attempting to RADAR identify the MYAM flight plan aircraft and issuing the IFR clearance I allowed the traffic situation to distract me and this prevented me from observing the traffic for the B712. I believe that I should have broken up the process for issuing the flight plan for aircraft landing MYAM into two parts (RADAR identification and clearance) allowing me to complete a traffic search for again for the B712.