I have just been given a 1996 Dodge Neon SOHC 117,000 miles It runs really well but the transmission is shifting really hard both up and down gears. I know that the computer module has been replaced. I have been looking on the internet for the past several hours but can't seem to find anything that points to a smoking gun.
If this is a automatic, I would start by checking your engine and transmission mounts. Have someone assist you, open the hood and start the car. Have someone shift the car to drive. If the enginge rocks while shifting into drive, reverse etc. Chances are the mounts are shot. Ive replaced many of these mounts. The only mount you can easily see is in the center front of the car, directly behind the rad support. I sugest using a flashlight to inspect the rubber. If this one is bad you should replace all three. If it is a 5 speed, I would check the clutch and release fork. 1997 & up neons use a modular clutch, I would suggest upgrading. I hope this helps. Good luck