How did my rear freewheel become incredibly tight overnight?
I decided to put up another answer because while Daniel's hunch and Billy's excellent answer get at the gist of the solution, I did find some specifics that might help someone else (though perhaps only a SRAM S27 owner).

First of all, the service manual is pretty helpful.

In my case, the "end cap" was tighening on its own during rides.  Normally, this little set screw is supposed to keep the end cap from threading or un-threading (ie. you get the adjustment just right, apply the set screw, and it should all just stay put).



The set screw actually applies pressure to a rubber o-ring, which in turn presses against the threads on the axle to keep the end cap from turning.

On my wheel, that o-ring had actually partially disintegrated where the set screw contacted it.  So instead of pressing a healthy o-ring against the threads, it just sort of mashed some rubber particulate in there.

The right solution to this problem is probably to get a new o-ring.  The route I took, however, was to rotate the o-ring about 90&deg; so the set screw was once again above a healthy bit of rubber.  I then re-adjusted the end cap tension and re-applied the set screw.

Hopefully that keeps me going for a while without tightening up again.  Thanks for all the help, everyone!