So, how important is it to participate in some kind of new-age/mind-body exercise? Am I just a stressed out nutcase who can't handle the relaxation of yoga or tai chi?

It is only as important to YOU as you want it to be.
Do you stretch what you think is enough on your own?

Personally I enjoy yoga. I am naturally flexible and it is the only time my brain stops racing around thinking about all the things I have left undone.
I try to go to a class once a week and I do a short video twice a week at home(with a small child crawling on me).
For ME it is important and very relaxing.

However, I would rather have dental surgery with no novocaine than go to a "step-aerobics" class.
I run a few times a week and enjoy it- but my mind is always going. I need the mind-off time.

My husband and I are ... obnoxiously in love and do everything together.

That is just nice. As long as there are no PDAs at the gym.

peace & relaxation
t