If the tuber is healthy it could certainly put out new sprouts. Its much easier to overwater them than under water them at the early stages. The tubers can rot, but if yours wasnt soft at all, then you have a good shot at re-growth.

I don't think that starting dahlias in zone 4 in April is late. Actually dahlias are native to Mexico, so they aren't cold weather plants and this season has been even cooler than average in the northeast if not everywhere, and dahlias don't grow very fast when it's cool outside. Plus in zone 4 you probably wouldn't want to plant them outside til late may anyway. I started mine in late April in Delaware, zone 7, and it was so cool here that they have still only grown to 5" tall so far! That and the crazy amount if rain has caused some tubers to rot and die. I tried growing mine in pots then transplanting them in the ground once they were over a foot tall and found that they don't seem to like being transplanted after establishing their roots in a pot, and the ones left in pots faired even worse. I Start my tubers either in the ground in mid to late May or let them sprout in pots them put them in the ground.

As far as whether your tubers will come back, I've had a few sprouting stems rot due to excessive rain this spring, but then I pulled the tubers from the ground and dried them off and replanted in drier soil and a few grew new stems.