I found a solution. (See last paragraph)
As mentioned before, I also think that the underlying issue is, that the game thinks that you still carry you sword. You can pick up any weapon only once, so it won't let you pick up a "new" sword.
Sadly, the mentioned trick of equipping and throwing a phantom sword did not work for me.
I tried cheats, but If a summon a new sword, I too can't pick it up.
I encountered the bug when I first met these snake plants, they threw away my sword and I could not pick it up again. I could also not pick up the next sword that I found. I reloaded to a savegame just before that, I still hat my sword, but once I threw it, I was unable to pick it up again, so maybe that bug is not so much related to those pesky plants. Now here is how to solve it:
Press Tab and type "cheatplease" to enable cheat mode. Press tab again and type "summon romansword" which will spawn the wrong sword, since i wanted to pick up a VikindShortSword. However, after equipping the roman sword, i can press 3 to switch to a club and if I then press 2 again I switch to the phantom Short sword I "had" the whole time. Now I throw away the roman sword, since I am no filthy cheater and continue playing with my appropriate sword. I suppose this fix generalized, just summon a weapon of that category that you do not have yet and try switching to the other ones. Once I reenabled my sword I could throw and pick it up again without any issues.
Here is an overview of all weapons:
https://steamcommunity.com/sharedfiles/filedetails/?id=398935934