Personally for me, when my calves get tight I tend to over-pronate. It’s not really about being aware of it when walking/running, it’s just that my ankle doesn’t have enough flexibilty so my body finds that flexibilty by collapsing the ankle inwards instead. But it can be caused by weakness as well, in the ankle and/or the hips. If you’re concerned, it’s worth seeing a physical therapist who should be able to help you figure out what causes the over-pronation for you personally.
New Balance do some of their models in 4E width (which I wear). Having said that, I think NB’s sizing runs very narrow in general, so 4E to me feels slightly wide rather than very wide.