Sounds like you're experiencing MIDI data "overload". This can be caused by MIDI data feedback, if you have both MIDI-in and MIDI-out connected, and your MIDI interface and/or software enable a MIDI Thru function.
If that's the problem, the usual cure is to turn "Local Control" on the synth "off". Unfortunately, the original DX7 didn't have a "Local off" function, although the DX7II does. If you can turn local off, try that. If not, you might have success by seeing if the MIDI interface or software allows the Thru function to be disabled. If that can't be done, you could try pulling the MIDI cable which is not being used at the time, depending on which way the data is going. That approach may or may not work if the application requires "handshaking".

--Barry