Sometimes, cats become anxious with a change in the household such a new family member or a move to a new home. However, it is also possible that the change in the household just uncovers a long term problem. Opening up a vacant guest room can lead to the discovery of the cat's use of the carpet in the room. A common cause of anxious urination is intimidation by another cat. Cat social signals are subtle and you may not know that one of your cats feels harassed.
Cats stare at each other for intimidation as well.
You may feel the assertive cat is just lying quietly but the body language of the anxious cat will give away the inent. An anxious cats slinks and increases distance between himself and the other cat. The anxious cat's tail will be held low and ears may be slightly back when the assertive cat is in view. The confident cat will appear to be resting and may even be purring. Be sure litter pans are clean and plentiful and easy to access.