I finally stumbled on my cleanest setup yet:

Tidy Cats Litter Box (see web site) with high back on the bottom and
hinged front on the top for easy access.  The high back means unine is
no longer being forced between the seam as it has on other covered
boxes.

Tidy Cats Jumbo Liner which the bottom fits into neatly leaving a
distinct lack of folds to catch urine and prevent it from reaching the
litter, while still being loose enough to allow a trick described
below.  A complete replacement of liner and litter is accomplished by
turning the liner inside out as one removes it from the box.

World's Best Cat Litter for its well known attributes.

I can peel clump off the back of the liner by carefully inserting the
scoop between the clump and the liner and flipping it forward in one
piece.

On the somewhat looser front, I can grab the liner where it emerges
from under the top and flip the clump off.  You can't do that with the
box, so that's why I like liners.

It's been about a week since I started this and the box looks fresher
that it or any other box has after a week.  I can give an update in a
few months to see how long it takes before I have to do a complete
replacement.

Now if I can just get one of my two cats to use it for something other
than urine.  He's been doing his other business usually in the same
small utility room as the box, sometimes in the hall outside.  I'd get
him his own box if I was sure it was from territorial issues with his
litter brother, who does use the box for all business.