Is it possible to change the content of the "watched" emails, so that they don't send the content of the post out by email?

It's going to take a modification of the handle_watchlist_notifications function in the libs/ubbthreads.inc.php script

Since this is a code mod, your best bet is to check over at UBBDev Instructions for the modification may already be posted over there.

Thanks, I'll take a wander over there.