After about two hours of work, this is as far as I got:
Note that the window that looks polished is the AWT GUI; the SWT window is the one above it.
(http://i125.photobucket.com/albums/p70/MrCamel7/bnubot/swt-gui.png)
http://rolyata.com/~andrew/code/bigbird/BrowserProject.java
wrote it like 2 years ago with SWT... probably won't help but eh?
Actually, the reason I was having so much trouble is because I didn't understand, at first, that only the thread which created the Display is allowed to edit anything. I've been making a fair amount of progress since then. It's still slow, since my motivation is so weak - after all, I've already got a fully functional Swing GUI.