My 2.4 swapped 98 R/T is doing this exact thing! I suspect the temp sensor is the problem, but my car hit a deer prior to me buying it so I have a few grounds that need replacing under the hood near the battery, so I'm not jumping to conclusions just yet.

 

Yes, I still have a sense of humor. How many "grounds" can there be and how do I, or anyone find and test them all? I intend to have not only the sensor, but the thermostat replaced when I have the system flushed. Will keep you posted(bad pun??)...

 

I fixed my R/Ts issue yesterday that was similar to yours, honestly for years my original Neon has been missing probably half of its grounds it should have and NEVER had any issues, the main ground that causes problems is going to be the one that is located on the top radiator support in front of the battery, unplugged/broke it will cause your lights to flicker and it will also cause your temp gauge to go nuts. Trace that one verify its condition and weather you need to fix it or not. Also check the starter wires that they are not corroded or hanging on by a thread or less.