ZDC Controller described a potential conflict event when two enroute aircraft were sharing the same transponder code resulting in incorrect altitude and associated target identification information.

Narrative

Air Carrier X came over from ZCC with data block saying he was at FL380 when he was suppose to be at FL360. I questioned the Controller at ZCC and he/she confirmed aircraft was at FL360. Air Carrier X checked on level at FL360 with his Data Block still showing FL380. I put him in at a reported altitude of FL360. When I did this another aircraft flashed Air Carrier ZZZ that was right on top of him showing FL380. Now during this time an aircraft at FL360 began to flash and he was traffic with the Air Carrier X. I asked the Air Carrier X if he observed an aircraft on TCAS converging with him and he said yes it was at FL340. This really set off alarms because I was showing them both at FL360. At around this time; I became aware of an intruder at FL360 already well into my airspace. I asked are there any aircraft on my frequency squawking XXXX the Air Carrier X said yes. That is when it finally became clear what was happening. The Air Carrier X at FL360 was sharing codes with Air Carrier Z at FL380 and the actual track for Air Carrier X was actually 20 miles ahead of his data block. This created much confusion because of the data block not associated with the actual track this put other aircraft in the vicinity of the actual track in danger. Recommendation; something needs to be done through automation to prevent a similar situation. This could have been tragic if I had descended the aircraft for his supposed traffic into another aircraft not realizing he wasn't where he indicated.