Here's what your relay should look like. The terminal marked "G" is the ground terminal and normally gets its ground from the neutral safety switch. Since yours has been hosed, someone must have provided another wire to this terminal. Check the wire going to this terminal or just substitute another wire from frame ground to this terminal and see if it starts. Might also check the other wires going to the relay for good crimps and connections.
First things first. You dont have any backup lights either? The 3 bare wires under the truck need to be fixed. You can get a replacement connector from Dodge #2926502...thay are not cheap($20). you might be able to find one at NAPA or CarQuest, I'm not sure. Or maybe a tranny shop? Your relay must have a reliable ground and the 3 wire harness provides this and the backup light function. You may need an assistant to work the start key while you feel the relay when your truck wont start. It must click in the start position. Also since I now know the 3 wires are not connected it will make no difference if you try to start in neutral. You could have other problems also but you need to fix the wiring problem you have alrerady.
Do this...take the wire off the "G" terminal on the relay...it should not start that way(just a test). Now make another wire about 2' long and put a slip on/spade connector on it. Place the connector and wire on the "G" terminal and run the other end of the 2' wire to a good ground on the body..preferably under a screw or bolt head. This will provide a good ground for the relay. We still need to determine whether the relay wiring is your problem and this will help.
Here's the spade connector type I'm talking about.
I forgot to give you a couple of other options. The 3 wire harness should be available at the salvage yard as these are a common item on Chryslers in the 70's and early 80's. Take your old harness off and bring it with you to compare, make sure any replacement harness you get is not already heat stressed. The harness may be available from Dodge also under part number 3730153. www.kollerdodgeparts still shows an internet price for them.
RamchargerCentral.Com is a Dodge truck community built by a vast number of experienced owners. Whatever you're thinking about doing from routine maintenance to full-on restoration, odds are somebody around here already has. Originally focused on the Ram Charger / Trail Duster, our members own, drive and wrench on Mopars from the 40s to the current day. It's not what you buy, it's what you build.