Remove paint that leaves mark on hands

I've just bought a flat and am in the process of redesigning and decorating. The previous owners have performed some shoddy paint work. The windowsills are especially bad. They've painted them in white but when you swipe your hand over the windowsill it leaves dry powdery white sticky marks on your hands. Note they painted it over 4 months ago. Can anyone advise how to remove this paint from the windowsill (or any other solution)?