I'll add one thing to this. If you do follow this route, find a new job and give notice to your company, be prepared for them to suddenly wake up, tell you they'll fix everything, make a counter-offer, and urge you to stay.

This doesn't always happen, but it has happened often enough to me, so you might think about the possibility in advance and what you would do.

For myself, I have NEVER taken the counter-offer and stayed. When I have had a bad manager, I regarded that as a failure of my manager's manager as well. I didn't think the situation would instantly be fixed, miraculously, if they did fire my boss or move me to a new position in the same company. But that is just my experience. It would be good if others who have had a different and successful experience staying would speak up too, so you can get the full picture of possible outcomes.