1998 Ford Ranger engine won't shut off

Suspect the starter relay first. Since this occurs while driving, the neutral safety switch will interrupt the circuit going to the starter relay, so, for example, if the ignition switch was failing, the neutral safety switch would prevent the starter relay from engaging. Somehow the smaller solenoid wire on the starter is getting 12 volts when it shouldn't. Other than something weird like two adjacent wires rubbed through and touching, a common suspect is rust buildup across the starter relay's contacts, and someone installed the wires on a replacement starter too close together, or they got tugged on, and the smaller one is touching the larger one.

To start with, swap the starter relay with a different one like it. Ford has finally started using a starter relay identical to what most other manufacturers have been using for many years, so you should have something else available to switch it with, like an AC compressor relay.