It's a bit hard to tell, but I notice that in the video you provided, the sounds shut-down sequence did appear to be operating pretty much normally the one time you turned the power off. That leads me to believe it's not a battery problem. It wouldn't hurt to charge it though, just to be on the safe side. Leave it on the track with power turned up and volume pot turned down/off overnight and see if that makes a difference. I suspect that it won't.
I still think trying it with another transformer (preferably an MTH Z1000) may do the trick.
