I just tried it by unplugging all of my connected devices and running only the displaylink adapter.
It worked for longer than it has since it originally stopped working - about five minutes. Switching USB ports, with everything unplugged, brought it back up for a couple of minutes, then it went out.
I'm kind of hopeful that a powered hub will do the trick and am going to run out and get one today. I'll post back with my progress.
Wish me luck - I really, really, really want this to work.