got any images of the mosses.
if they came from nearby areas then i would check out those areas to see what the conditions are like there.
two different types may mean two different approaches.
main thing is to maintain sufficient humidity. you need to use rain water, distilled water or r/o water and mist the surface of the moss. come back in about 4 hours and touch it to see how dry it has become.
sounds like the moss on the sand may prefer it wetter than the other but it is hard to tell without seeing what they are.
The spongy type moss sounds like Leucobryum but that is just wild guess!!

Here is a link that might be useful: Leucobryum

Thank you terrestrial_man. Why can't I just use the hose?

Does your water supplier mail water quality reports to its customers?
Is your water straight out of the tap?
What is the pH of your water?
Do you have plants in pots?
If so does the surface of the soil in those plants appear to be crusty or have surface deposits?

Mosses are subject to salt burn even at low amounts. This may explain why they tend to occur on very poor soils. This also can be related to the fact of just how simply formed they are. While some species have adaptative features that allow for survival in desert settings, most of the mosses that you think of as being mossy are generally found in damp to wet places and are probably not protected by any sophicated system to prevent dessication and death or even to protect the cells of the moss from becoming dried out by the presence of salts on the surface of the leaves which causes the water of the moss to move out of the leaf which causes cellular death.
While hard waters can be destructive to mosses I would think that any water with a high enough level of salts in suspension may be harmful. pH levels would tend to indicate that certain of the salts in suspension would be held in suspension due to ionic polarity while others would precipitate out for the same reason. For this reason some moss species are not affected.
One methodology I learned years ago while working at a Santa Barbara orchid grower is the practice of leaching out. That is watering excessively so that whatever salt build up begins to accumulate would be washed off though it would be replaced by a fresher accumulation. With daily drenching the aim was to keep the salts in suspension or so temporarily held by the soil and potting media so as to have no impact upon the roots of the orchids. Applying this to mosses it would make sense when encountering those species that occur in very damps situations. Thereby through inference I would presume that such encountered species would be more imperiled by the use of tap water than not. That is why knowing the habitat where your mosses came from is a good foundation for basing successful culture for growing them.

This is helpful. I will do some research,
- Thank you.