Thanks, Jeremy - I followed the notes you've sent and it's currently working fine for me. I did find that rearranging my widgets temporarily switched it back to the "My Tweets" broken link, but I was able to paste in the correct widget ID code, save it successfully, and it's back to normal.
Not sure about the glitchiness, but I'll watch carefully to see if I run into any other problems and report back. Of course I'll also make sure all of my wordpress sites have Jetpack up to date as I switch out the rest of our twitter widgets.
I don't know how much difference it would make for anything, but all my sites are running on WIMP servers - it generally doesn't cause any problems for this kind of stuff, but it's the only major difference in our configuration from most people that I can think of.