My wife has a 2001 Stratus. The starter bendix gear was still turning after the engine was started. Being as the starter and solenoid are one unit on this model, I replaced the starter with a rebuilt starter. Now I have the same problem but I now have a significant momentary grinding noise after the engine starts. This is not constant. It only happens every few starts. I assume I just got another bad starter. Any suggestions?

Yeah, you could have another defective starter. It's not your wiring b/c it's starting up just fine; i.e. the battery is supplying enough juice to energize the solenoid.

Info for all! The problem is fixed! After installing a new starter and having the same symptoms again, I happened to turn the car off, then turn the key to the "accesory" position to roll up the windows, and the car started! Then I started troubleshooting and found that the car stated in the accessory position 9 out of ten times! The problem was a broken ignition, not a defective starter! When starting the car in the usual manner the ignition contacts were sticking in the "made" position causing the started gear to remain in contact with the flywheel! A new ignition fixed the problem. By the way..the 2001 Stratus has 5 possible ignitions.
Hope this help all others with this problem!