The solution is...use fewer dishes!
Thank you very much, it only took me more than 2 years of complaining to think of it. (I say two years because, while I've done dishes by hand for over 5 years now, I've done significantly more cooking since marrying DH (hence more pots to clean) and lets not forget the million bowls he uses.)
The best part of this solution is the implementation strategy. How to use fewer dishes? The answer: Get rid of all but two bowls, plates, and sets of silverware! Yes. This might work. ("Get rid of" can be loosely interpreted, of course. My version of "get rid of" is to put extra dishes in a slightly inaccessible cabinet with my China plates.) Now we will not be tempted to grab the clean bowl instead of cleaning the one in the sink, because there will be no other choice.
Let us cross fingers now.