Thanks for the reply. I just bought this truck Saturday. Prior to that it worked fine in my old truck that has a Pioneer stereo with Bluetooth. There were occasions the Bluetooth would get flaky, dropping and reconnecting repeatedly during calls, but a phone restart fixed it. I never experienced the behavior I have now where it drops Bluetooth disconnected from the charger or with the screen off. Also, I used it with my wife's car Saturday with no issues as well.
Even so, I'm inclined to think the issue is still with the phone. Unplugging the phone from a charger and the screen being off or on should be completely transparent to the truck. There has to be something in the phone that affects Bluetooth when the charger is disconnected or the screen turns off. How else would the truck know that those changes occurred with the phone?
I need to try my wife's iPhone with the truck, they may give me a better idea also.
My phone just did a system update this morning. I'm going to clear the cache again. Also, I've read that I may need to reverse pair the phone. Instead of looking for Sync on the phone I can look for the phone on Sync and pair that way.