Sounds like the thermostat

 

If the car was running, and the temperature gauge was reading hot, and the fan was not on, you've got a fan relay issue. Hot wire the fan directly to the battery and see if it comes on that way (left side of the connection is ground). If it works that way, the fan isn't bad and I'd recommend replacing the ECT switch.

If the '93 DX is the same as the '93 Si, it's the brass connector that's screwed into the thermostat housing. Since you'll be in there, you might as well replace the thermostat too.

Good luck.