Starter stays engaged after engine is running?

Correction: I changed the "micro" relay associated with the starter. Next move would be the Bosch style mini load relay, but I doubt that's bad. I'll pull the PCM and test the green/orange lead on the C3 PCM plug as you described. Back of my old noggin tells me the PCM may be the culprit. I won't get back to you for a few days. I do thank you for sharing this info with me.
Note: I thought sure the ignition switch module was the problem. I had it in my mind that this switch just had a case dirty wipers, like and kind to the old-style intake AFMs. Not to be.

Re your previous reco on trouble shooting. I was a bit stumped initially because I couldn't find the dark green / orange wire. My Eagle Eyes. What I did find was some mickey mouse wire repairs near the loom connector to the black ECU plug. They were buried under suspicious electrical tape cover repairs. I made quality solder repairs w/ shrink tubing at the defective connections. 1st two attempts at starting the engine were perfect. The 3rd and 4th attempt. Starter run-on. My ECU was under warranty, so I had a replacement in hand. Swapped the ECU and all is well. Ten attempts at starting over a 3-hour period, hot/cold engine. All is well so far. Note: this 2008 300 car was an Enterprise purchase in 2010 w/ 17k on the counter. The wire repairs are a mystery. You never know what you're going to find. And of course, knock on wood because it's a Chrysler product. Beautiful car in impeccable condition. I sincerely thank you guys for your detailed recommendations.

A new day. The same problem with starter run on, only worse. Back to the wiring.

It's been a while. I wanted to get back to you guys with the final resolution of the 08 Chrysler 300 dilemma and what I learned about my issue with starter run-on once the engine was running. Out of the blue the starting issue went from starter run-on to the engine won't start. My check of the injection fuel pressure showed that the required fuel pressure was not being met during crank/initial start-up. Once running no fuel pressure issue. I swapped out the pump. Here it is weeks later, and no starter run on, and it starts perfectly every time.
My question is, is the ECU circuitry so complex as to require observed fuel pressure during cranking (and initial start) and in turn keep the starter engaged until that optimum fuel pressure requirement is within operational tolerance? I mean I'm happy and my wife is happier, but dang! If my assumption is correct re the complex integration of sensory input to the ECU, then this is a definite reliability issue that Uncle Chrysler needs to resolve. And, if my assumption is correct, this should have a specific trouble code. Dang! Let me know what you think. And thanks again for your knowledge and skills. Regards from the old and more arthritic Keith