I've gone to the bar diapered, and yeah, I've soaked the one I was wearing. My solution was pretty simple though. Take it off. I had my fun, and since I don't need to wear, I have that luxury. I ran out to my car, grabbed a few wipes from the glove compartment, then went into the bathroom and just threw it out.
I know it's not quite as fun as staying diapered, but in a way, it was a bit of a relief once I wasn't wearing it any more.