You and everyone else... heck of a thing, at least it's not absurdly expensive...
I think you do need to send them another transcript, for sure at some point, might as well just bite the bullet and send it now. I'm honestly not sure if that'll cause more delays. Logically, I wouldn't think it would, except that everything causes more delays...
You might also be able to get away with waiting until the end of your summer session (so that the classes will be graded), especially if it's one of those short jobs that get done mid June.
How's that for waffles? I guess the bottom line is, it probably won't hurt to send an updated one now, but if your class is short, you can probably kill two birds with one stone and send the transcript once it's in the books.
This is not always wise. I know that at USF they don't process any summer grades into official transcripts until afte September. So you'll be late. send a letter of update to schools individually with a grade sheet at a later date.