Tiger lilies are pretty resilient, but they do use the stem to feed the bulb for next season. I'd just dig a hole 6" deep and plant them, even the sorry looking one, again - nothing to lose at this point. If there's any stem at all left, leave it attached.

I had a similar tragedy and have three very large bulbs that I dug up yesterday. Is there any hope that they will come back this year, or should I just replant them and hope for next year? Also, now that I have the bulbs out and see that they're quite large, is there a way to propagate more lillies by dividing the bulbs?

No, they won't come back this year. Just replant and hope for the best for next season. Replant as is - don't cut the bulbs apart. (Lilies can be propagated by scaling the bulbs, but you'd have a few years to wait before seeing any kind of results.) For me, it takes @ three years for the little bulbils that form in the leaf axils of Tiger Lilies to reach blooming size.