...since I reinstalled Windows and the Steam Client. A lot of other glitches that I used to have issues with are gone, but now I get no Dragon Souls whenever I kill a dragon. It doesn't matter if it's in Skyrim or at Saltsheim.
It bothers me because I use Dragon Souls for other things than unlocking Shouts: I have a mod called 'Use Dragon Souls for Perks' and have done a lot of work in order to plan the whole thing.
If anybody knows how to fix this issue I'll be very grateful.