Odd ignition and electrical problem

You likely either have some wiring hack someone did that is causing some sort of backfeed, or more likely some sort of cross between wires in the harness (such as melted from a previous short) that is causing this.

If you do not have one go to MyMopar.com and download a free service manual. You may have to accept Dodge vs Plymouth, etc.

You likely are going to have to pull the harness out or at least part way, unwrap it and inspect for overheated / melted insulation, etc. Before you do that, get under the dash, and in the engine room, with a light and examine the various connectors for melt and overheating