I got sent by the Greybeards to find the word of power in Archwind Point. There I got the third word for Drain Vitality. However the objective didn't disappear and I can't get the locations of any new word walls from the Greybeards.

Does this have something to do with it being a Dawnguard shout? Is there any way to fix this?

It's a known Dawnguard bug. No way to fix it other than reloading a save before you got that quest.