Across multiple models but with broad nibs. Usually it leaks into the cap when closed and then the part of the pen the closest to the nib is covered with ink.

I need to take a look at this. Are you sure that you're not gripping from the lip which locks and creates an air-tight seal? The grip would make it very uncomfortable though.

I have a medium Lamy which writes broad-ish (because of production tolerance + well smoothed nib due to long writing sessions) and wets its sealing lip, but even that one never stained my hand.

Which ink are you using? Is it extremely wet and eager to flow? Do you fly with the pen? Temperature around the pen?