If they're inundated, then yes, they will rot.

I've had similar problems with water and low lying areas.

Believe me, I can tell you about low lying areas..

The best thing I can say is hope the ground soaks up some of the water, or it evaporates soon. Does the soil contain too much clay -- too compact?

If they're really important you can try pulling them and the soil around them up, then transplanting in an area with a slightly higher elevation [ mounds of dirt, for example ].