https://bugs.winehq.org/show_bug.cgi?id=30981
--- Comment #17 from Vlad vblanton@gmail.com --- This bug is present in windows as well. I search the internet for "skyrim keys get stuck keep moving" and found a number of posts. for example, this steamcommunity post: https://steamcommunity.com/app/72850/discussions/0/617335934140598491/
for me, the issue was with repeat keys. normally, if you hold a key down when you are typing anywhere, it will keep printing it out. skyrim seem to have an issue with this.
I use Gnome, and my solution was to turn off repeat keys in gnome settings before I played and to turn it back on after.