Fortunately, I was able to find these tubs at - I think it was Dollar General - which work very well and stack nicely. I found an absorbent drain mat at WalMart, too. I was originally planning to put it under the dish drain, but then discovered I could simply put the dishes on the mat to drain.
Then, when I'm done, I replace the sink cover over the left sink.
Then I slide the mat, dishes and all, onto that cover. Drop the extension and the door is once again usable.
Added bonus: the drain mat goes in the washer when it gets dirty.