Thank you all for the advices. Using another lens did show that it was the sensor.
I tried blowing the dust and it did indeed remove some of it. But other appeared, it seems it has a lot of dust in the body, and blowing it only "moved" the dust around. And some 'fuzzy spots' wouldn't go away at all, I'd bet it's some kind of droplet that got on there too.
So right now it's at some local photography shop for a full clean-up. I know, I know, it's supposed to be easy but I don't have the time or patience to order the kit, try to clean it myself, fail, retry it, until finally getting it right
So it'll cost me (around 50€, so that's like 56$), but I don't have to do it myself.
I'll try setting the dust-cleaner-on-startup thing when I get my K50 back too.