Wait! I'm confused. The cord on your needles broke, right? But there's nothing wrong with the sweater itself, correct? Well, okay, except maybe for some dropped stitches?
So why not grab a darning needle and a long piece of yarn and string all your stitches on that? Then when you get the new needles (and if they pass muster), you can slip the stitches onto your new needles and take off again.
I most definitely wouldn't frog it, though, just because of a problem with the needles (which can be replaced). Now if you'd screwed up the pattern or had lots of incorrect stitches, then, yes, frogging it might be in order.
At any rate, I'm sorry to hear about your frustrating experience!!