Thanks.
When I deactivate the option you show below, it does allow me to choose the shipping method via an unattractive, secondary screen.
I am using version Cleanmart version 3.2, however, and I'm only using weight_countries shipping methods without any 3rd party shipping rate plugins. In fact, I've tried deleting all my shipping rules, adding only a single test rule for free shipping using weight_countries, and I still have the same problem unless I deactivate the AJAX feature. (My site needs a dozen weight_countries shipping rules to work correctly.)
Can you think of anything else that could be causing this issue? I even tried deploying the jqueryeasy hack, and that didn't help, either.