I'm going to plan on bringing some dry-goods as some food items. And hand-sanitizer!
I'm new to the living-history camping thing too, except for my trips to Pennsic. But I've also done extended trail hikes.
As a general rule, I try to touch absolutely nothing in the porta-potties with my hands if I can at all help it. If I need to flip the seat up, I do it with my foot. Open the door with my elbow, that sort of thing. And of course using hand-sanitizer back at your tent might be a good idea afterward.
Cleaning your dishes is a non-issue, I think. As long as you rinse it after each use, it's pretty good. That's all I'd do out on multi-day hikes on the trail. As others have pointed out, if you use soap but don't get it all off, you're going to have intestinal issues. I think it's better to just give it a good rinse so there's nothing left on it to decompose. It won't leave any meaningful amount of anything on it to be an issue.
Just keep an eye on your feast gear, or don't leave it where passing visitors will put their grubby hands all over it.