They'll keep growing until they die. In the tropics they are a perennial, with people eating the leaves. I've never tried that.
You absolutely want them dug before a frost. They are very sensitive to freezing. If it frosts on them the tops die immediately and the frost travels into the potato, causing it to rot. The will not store at all if the frost gets into them. If you are stuck where you cannot dig them before a frost, at least cut the tops off of them before the frost. That helps some.
I generally watch the weather reports and wait until a few days before a frost is possible before I dig mine, but I'm up for gall bladder surgery in a few days and don't know when I'll be able to dig them, so I went ahead and got it done.
Cattle, horses, and pigs really love sweet potato vines, but don't give them so much that they get sick, especially horses.