If I’m not mistaken, Rebbitzen Soloveitchik did cover her hair in public.
The fact that a good shaitel often looks better than one’s own hair is irrelevant – thre is no inyan that a woman shouldn’t look attractive. The issue is one’s own lliving hair, which is sensuous. Rabbi Soloveitchik absolutely said that married women were required to cover their hair in public.
There are many nice ways to cover your hair – be creative, and think positively about it, and hopefully it will become more pleasant for you.