I wouldn't buy a relay at this point, you've swapped them around and still have the problem so it doesn't sound like it's a relay.
When you have the no start condition you'll need to verify that you have fuel pressure using a gauge at the shrader valve on the fuel rail. Pressure should be over 40psi.
Next you'll need to verify that your coil is providing spark. Pull one spark plug up a little and listen for the ticking produced when the spark is jumping from end of the wire to the spark plug tube. Pull it all the way out and stick a screw driver in there and test it in that manor if your prefer.
If you have spark, verify that the injectors are spraying. One wire has battery
voltage on it and the other leads back to the computer. The wire that has the battery voltage is shared with the other injectors. I beleive it's green with an orange tracer but I don't have my manuals with me nor do I have my neon with me at the moment. The computer provides the ground for each injector seperately so the ground wires will all be different colors. If you have a "noid light" then you can just plug it in and look for the light to flash as you're attempting to crank the engine.
At the coil there's three wires. One is battery voltage while cranking and running. The other two are "Coil Drivers" which are grounds that the computer applies and takes away. Once you have verified that you have battery voltage then you can plug in your noid light to one driver wire and the battery voltage wire. It should blink while you're trying to crank it to show that the computer is applying and removing ground in a pattern. Move the noid light to the other ground wire and reuse the power wire to make sure both circuits are working.
If you have power on the injector wire that's common to all and power on the wire at the coil but the noid light does not illuminate then your computer is not providing the necessary grounds for each component to do it's job.
You can usually find PCMs (Powertrain Control Module, Chrysler's version of an engine computer) at a junk yard and they're easy to swap out. If you diagnosis leads you to the computer, consider upgrading to a Mopar Performance Computer or another performance one. All will require premium fuel but you'll get a little more umph out of it and I always prefer to upgrade a part once I've found it to have failed.
Hopefully that will be enough information to get you working on it today. I will not be able to respond back until late tonight but I will be back in touch soon.