Does GEL-LAY needs to stay moist?
No, GEL-LAY should not be kept moist to stay flexible. It is made from TPE and filled with small PVA particles. It is advised to print less than normal amount of walls and infill so that the PVA can be rinsed out more easily. 

If you look at the way the manufacturer rinses the material, you frequently see ultrasonic baths that improve the dissolving of the PVA. Once the PVA is removed, TPE is all that it left, which is elastic. If your print is still hard after drying it means that not all the PVA has dissolved yet, once dry it will become gluey/sticky again like PVA, hence the cardboard structure.