The phrase is old-fashioned; use it to sound dramatic or humorous in modern settings. He stormed out in high dudgeon, as if he were a character from an old novel.
Pro Tip 2/3
Strong Emotion
Use when describing a reaction filled with intense indignation or resentment. Upon being falsely accused, Maria departed in high dudgeon.
Pro Tip 3/3
Public Display
Often implies someone showing their anger openly, especially where others can see. At the public forum, he left in high dudgeon, making sure everyone noticed.