Do you have the passive alarm system on yours? I have a 95 Mystique LS and it has the passive alarm system. After 30 seconds to one minute it automatically activates the starter disable. You can either turn off the alarm, which on mine is having the key in the ignition in the start position and pushing the red led lighted button under the steering wheel.......or after 30 seconds to one minute I press unlock at which time the car honks telling me it's now disarmed and the car will start...........
If this isn't your problem.....then I don't know what to tell you.
Good Luck...
Ryan