I was thinking about your dim bulb issue, and looking over the wiring diagram I have for the 2005 Patriot, unfortunately it doesn't show where the dash turn signal indicators tie in to the circuit (they don't even have a detailed diagram for the turn signals). But we can still try to divide the circuit and eliminate a few things. You want to unplug the Chime module to see if that stops the dim bulb. If not, plug it back in and then pull the turn signal flasher, see if that shuts off the dim bulb. Also you can pull the stop/turn bulbs from front and rear (check and clean those connections).
Usually when one bulb glows dim (on a bulb that should not be On) it is indicating that you have a bad ground somewhere, and the circuit is "back feeding" to ground through that dim bulb. There are a lot of places for a bad ground, so pulling the flasher or chime is a good starting point to start tracking the path (note that if removing the part stops the dim bulb that doesn't mean that part was bad - it may just be in the path the circuit is using to get to a ground). But it will help narrow it down.
I color coded the brake/turn signal diagram to show the normal patch of power flow, but without knowing where the dash indicator lights tie in, it will take some exploratory work to figure out.