If the moss is grown in water then it is used to growing in water not air. If it is moss collected from the wilds on trees or other areas not under water, or purchased from vendors supplying from a forest, it would grow better in air than in water. It depends upon what it is used to.
To make the transition take either version and place it where it grew, either air or water, but just at the edge so it will need to grow up and out of the water line, or down into the water. It needs to have something to grow onto - say terrestrial moss to get to grow into the water, on say a chunk of bark dipped down into the water.
If you are using water moss, it needs to be placed similarly so that it will be under water with the bark going out of the water, and in time it will likely grow up and out onto the moist bark.
Also as a side note, some kinds of plants do not live out of the water environment and visa versa.
here is a bit on moss
To attach moss to DW for under water, use aquatic moss.
You can use a little drop of glue 100% cyanoacrylate usually super glue gel. Leaves a white film, so be artful and hide the sticky part if you use that. Can add a bit of ground bark to the remaining sticky part to cover up.
100% Cotton thread in color coordinated variety, rinsed in water.
Wire twist ties, color coordinated zip ties, or rubber bands can work, remove when grown in.
All the actual part to grow into the wood and attach takes time.
Out of the water, similar thing but should use moss grown above ground to start with.