Signal lights won't blink, simply stay on

First, the tail circuit is completely separate from signal, and the hazards have a separate flasher from the turn signals, as well as a different power source, as they are powered all the time. The signals may have a bad flasher, blown fuse, etc You can download a factory service manual for free, from MyMopar.com.

You say the side markers work, does that include the rear? If so, this is starting to sound like a bulb/ socket problem