If needed, keep changing the paper every couple of days until the pressing is quite dry and flat. Remove the protective fabric layer and make sure the collection details and seaweed identification are recorded with the paper. You now have a pressing of a seaweed. Many varieties of seaweed will actually have bonded with the paper underneath. Preservation One of the reasons you soaked your seaweed in fresh water was to rinse away the salt.
Salt crystals tend to absorb moisture from the air.
If your specimen become even slightly moist it will be vulnerable to fungal attack, so try and keep your pressings dry. Cover the specimen with thin fabric, sponge it off, then press and dry with newspaper. You should also keep them out of strong light as this will cause their colours to fade. Indeed, to keep your pressings in the best condition, store them somewhere cool, dry, and dark, and maybe with some naphthalene flakes at the bottom of the box to keep insects away. Afterall, you can never spend too much time down by the sea.