Yes, he "covered" it up, but he forgot to wipe away the chalk mark on the wall indicating that he had been there searching the tower. I'm interested to see where that goes.
I agree that this could definitely fester and potentially destroy Adolin. Reading about the lies that Sadeas was going to spread about Dalinar made my blood boil, but I think that Adolin's actions were too rash. I'm not sure if killing the man is going to stop the force that he was, mainly because his wife is so involved as well. There are worse men than Sadeas.