After sleeping on your question; I finally realized what is going on here. The sticking part is called an EGR (exhaust gas recirculation) valve, and it is sticking partly open, instead of closing completely. The purpose of this valve is to recycle a small amount of exhaust back into the motor while the car drives; but this valve normally closes when the motor idles. This reduces emissions of nitrogen oxide; which is a major pollutant. When the EGR valve sticks open; it lets more exhaust gas go past the throttle and into the engine. This will increase the idle speed; just like if the throttle was sticking; but it is not the throttle which is causing this, it is just the EGR valve. When the EGR valve sticks partly open; the engine's idle speed is always too fast; but when the transmission is in drive; it is more difficult to notice because of the load the transmission puts on the engine. However; when the transmission is shifted into neutral; there is no load placed on the engine; so the idle speed becomes noticeably faster than when it is in drive.
The way to fix this problem is to either remove and clean the EGR valve; or just replace the EGR valve. This can be done by an independent shop; and probably will not take much time. Here's a link to photos of the EGR vave for a 2001 Chrysler 300M with a 3.5 liter V-6 motor: http://www.rockauto.com/catalog/moreinf
o... This part sells for $172.79 from www.rockauto.com The dealership will probably sell the equivalent part for several times that cost.
Here's a link to photos of the EGR valve from a 2005 Chrysler 300; with a 5.7 liter Hemi V-8 motor: http://www.rockauto.com/catalog/moreinf
o... This part sells for $57.99 from Rock Auto online.
I suggest you phone some local independent shops, and get estimates for replacing or for cleaning the EGR valve on this model. You'll need to tell them the year of your car, and the engine size that is in it. There is a label in the engine compartment (usually on the underside of the hood, or near the radiator or the fender), which lists the emission control information for this car. It includes the engine family; which shows the engine size if someone knows how to decipher the letters and numbers. Just copy the information from the top of this label; and if asked; repeat it to the person at the shop. They may be able to know the engine size just from the model and year; but in some years; the Chrysler 300 was available with 5 different engines.
If you don't know of a trustworthy independent shop in your area; I can recommend some. Just click the "answer this question" button below this message; and type your postal zip code in the box that appears. Then click the "submit answer" button. I'll try to recommend some good shops.
I'm sorry that I was initially confused by your report; but I'm glad I finally figured it out.