300M Starter clicks... now wouldn't turn off!

I have had a problem with the starter in my '99 Chrysler 300M clicking a few times before starting (one click with each key turn). This has gone on for about 2 months and I had come to the conclusion that it was either dirty connections or a bad starter. I had planned on testing it at the starter during one of its cold starting mornings until this morning...
I was going hunting and was loading up to take off in my '02 Grand Cherokee Laredo when my wife was going to work (4:45am...CoLd). The 300M was acting up as usual so I decided to get in and try it myself. Instead of trying the repeated clicks, I held the second attempt for about 3 seconds. Then on the third turn, the car started right up. BUT my starter kept running and wouldn't shut off. Even when we shut the car down it still ran. We turned the ignition back on engaging the motor again, backed it up a foot or so and tried to shut it down again...no change. I ended up running around the car frantically trying to find a fuse for the starter and couldn't quick enough so I tried to turn it off again. This time I let the engine turn off for a few seconds and when I turned the ignition back on, the starter stopped. Talked about a frantic morning (killed my second wall-hanger whitetail in 6 days though-made up for it)!
What might be the cause of this? Does this information lead to a certain conclusion about the starter?
Thanks for all your help and time either way.

the fact that the starter is turning says the starter itself is OK, the problem appears to be with the solinoid, some troubleshooting with a meter will verify, there is a large wire from the battt positive to one side of the sol, the other side of the sol connects to the starter, when you turn the key 12v is supplied to the sol on a small wire, this causes the sol to pick supplying 12v to the starter via contacts inside the sol, sounds like your sol hung

yup replace the starter...