ZJX Controller described a stressful event when traffic at FL340 and traffic immediately below at FL320 swapped Mode-C information; indicating a climb initiated by the aircraft at FL320; the reporter indicating Mode-C swaps are problem.

Narrative

I had a Lear Jet level at FL320 and he was looking for higher. There was an air carrier directly above the Lear at FL340. It was a busy sector with aircraft deviating for weather and I was trying to climb and descend aircraft at the same time. All of a sudden the Lear Jet showed an altitude of FL335 and I thought the aircraft was climbing; so I asked him to say altitude and he did not respond. I then tried to turn the air carrier aircraft 50 degrees to the right but he would not turn due to weather so I climbed him to FL360. I then went back to the Lear Jet and asked if he was level at FL320 and he reported level. It ended up being a Mode-C swap. I thought I was going to have a mid air collision. It was a terrible time for a Mode-C swap to happen considering the traffic volume and unusual situations going on at the sector. I strongly believe that Mode-C's transposing is a huge factor especially in my situation. I feel this problem somehow needs to be addressed. This situation could have escalated and made a worse problem.