Adolin is actually in a different Chain of Command from Dalinar now. Adolin comes under Queen Jashna. I'm not really sure how that's works out. Since Dalinar is King of Urithiru. Also since he is a High Prince now. He is the General of the Kohlin Army. It would be his right to change any standard he wants. He may have to follow orders of the unified forces. Usually personal standards are left to the individual countries. You could use NATO and any member nation for example.
ALso I would like to dispel something I read or heard. Someone said Adolin and Shallan might be interested in twist with Kaladan. Adolin want even get something going with Vale or Radiant. How could we expect him to share Shalan with someone else.