Update: On the way home today, the weather was perfectly dry. Again, the computer worked for roughly a mile, and then it started going crazy again, typically registering somewhere between 60 and 90 mph. So I believe this effectively rules out wetness as the cause.

Right now I have no idea what is wrong. I'm going to try moving it higher on the fork, I think, but I still don't know how the sensor should be oriented for maximum effectiveness. If it still acts up after moving it, I'm going to return it.