I just got an e-mail from Lucas at Apogee:
"Just an update on the status of the switches. We located a more heat resistant version and they are on order. So probably somewhere next week we'll have them. I'll let you know when they're in and we can set you up for a replacement."
In case anybody wants this route, now you know they have dealt with the issue. I think the clue was that the Rosettas use the same switch without problem, but do not get nearly as hot as the ADA16Xs, so heat resistance was the obvious answer.
Goddamn those units get hot! Some op amps are being run class A, as I learned, and that makes it even hotter.