Black Water backing up in shower when driving.

All the truck stops and Flying J type stations along the highway have dump stations.

I'd probably search out one of the lists of dump stations and use one of those to dump before going home. You'll be surprised at how many dump stations that are out there for free or a nominal price.

I'm sure. This TC only has one waste tank(38 gallon). Tank wasn't completely full. In fact I would be surprised if it was even 1/2 full on this trip. I only had one TC shower, with limited sink water use. We have taken several trips with this TC and this was the first time it has happened.

Flat rubber stopper over drain with a weight on top?

Are you sure it was black water, rather than gray? Shower and sinks are supposed to go to gray tanks, not black.

Only solution that comes to my mind is take shorter showers so the tank is not as full. Or dump more often. Backup into shower when the tank gets too full is a design "feature." Never seen one back up into a sink, but it evidently happened to you.