clarks has a write up on testing the DME and symptoms of failure.
DME Relay Information and Testing
The relay also provides power to the DME, so it could be causing your problems. I'm kinda doubtful, but the relays are cheap enough and you should have a spare in your glovebox anyway, so you might as well order one. I'd check your wires and grounds too. Mine did something like that because some idiot (me) didn't have the negative battery terminal tight enough. It was tight, but just not tight enough. These cars are really sensitive to ground/electrical issues. That fix is cheap and easy, before you tear into something more expensive and time consuming.