Hey, just read the last answers after months, I forgot about this post.
I never really managed to get the intonation right that time, but I now got new strings (D'Addario Balanced Tension 50-120) and did a full set up, including intonation, and it seems to be fine. I think I know thought what was wrong then. I believe all my saddles were way too close to the neck (almost fully extended). I pull them all back toward more or less half of their length and work from there, and they were almost all perfect from the beginning.
is it possible that fully extended saddles (towards the neck) can cause this problem, maybe by moving too much when vibrating?