Does GEL-LAY needs to stay moist? POROLAY GEL-LAY is a kind of jelly-like material made by exotic filament producer Kay Parthy / CC Products.

It is a mix of TPE and PVA that is supposed to have a jelly-like feeling. Once printed, you need to let your print soak in water for several days to remove all of the PVA, then what's left should have a jelly-like feeling since the PVA will have left many micro-cavities in the object.

However, my initial tests end up producing prints that feel a bit like jelly when wet, but reverts to a sort of cardboard feeling once dry.

Does this material needs to stay moist to keep the jelly feeling or not?
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.