I don’t know anything about pressing them, but the ones we’ve used for teaching purposes in our botany labs have all been preserved in liquid, e.g. formalin or ethanol/glycerol. Apparently that’s also the first step in pressing them - found some info/instructions here: https://naturalhistory.si.edu/research/botany/research/algae/algae-preservation-techniques
You can certainly press seaweeds, the same way you would press plants.
(If the species you’re working with tend you stick to the newspaper, you can press them onto archival paper, which can then be glued to the mounting paper.)
I dried them up on thick paper which can absorp water…but do not make it to wet. Or put paper in a bord, and then some water and the seaweed…and pour the water of and let it dry
Take a flat bord, schaal, put a white paper on the bottom and than 2mm water in it. Put then the chara, seaweed or kranswier in it and stretch it.
Get the water of by pouring or else…
Chara are put on paper without drying as far as i have sen.
There is a pdf describing one preservation process downloadable here: https://spnhc.org/resources/how-to-prepare-seaweed-specimens/
This is also how I was taught to do it - pretty much just floating the seaweed in a bowl of water and then placing thick paper underneath it, slowly lifting it up then splaying out the fronds with a utensil. It seemed very effective on smaller specimens I’ve seen/tried.
Not really what your are asking, but once I collected seaweed to Id them based on morphology with microscopes for my botany course.
The idea was to collect one day and inspect them days later, so as a way to preserve the characters seaweeds were put in plastic bags with seawater and then frozen. This way their cells wouldn’t burst or anything like that. Sadly we didn’t get to see them in the lab because the pandemic started.
Due to their high agar content most seaweeds can simply be floated onto the page and left to dry. This is the way that the WA Herbarium collects seaweeds but thick paper must be used.