I have a 2003 T&C 3.6L with tow package, purchased used (privately) with 25K miles in 2006. Since I have owned it, it will occasionally not go into any gear when first started after sitting at least overnight- basically stays in neutral until engine is revved to about 2500 RPM. You can rev it in neutral or while it’s in gear. This always works, and the transmission will then work fine until another overnight stay. I have had the transmission serviced 3-4 times (once when it first happened, and then at recommended intervals. The car now has 96K on it, and it has been happening less frequently (in fact, it hasn’t occurred in about two weeks). I have discussed this with two Chrysler dealers, and have left it overnight twice, but of course it didn’t repeat at the dealer. I haven’t been able to correlate when it happens with anything except: it seems to occur more frequently when garaged in Pittsburgh than when sitting outside in Myrtle Beach (I spend time in both places).
@Transman will probably know the exact answer but I think you have a problem with your torque converter. It’s acting like a “stall torque converter” used in racing. The problem may be mechanical or electrical in nature.
Don’t know the answer to your trans problem but you don’t have a 3.6 engine, 3.3 or 3.8 were the only choices.
My typo - engine is 3.8L.