Polish, but rarely, and use much less polish than you would for calf. In the meantime, buff with a little water to bring the shine back.
Why? I polish every pair of shoes after each wearing. Including cordovan. No problem. I have had black Alden cordovans for about 5 years now of regular wear. They have been professionally re-soled I wear them so much. And they look brand new. There are literally no creases, not even the "rippled" effect that you generally see with cordovan. My father has some calf shoes that he has worn for 20-25 years, after polishing every time he wears them. Regular polishing keeps the leather healthy and strong, prevents cracking that results from little bits of dirt and dust that you can't even see cutitng into the leather. I simply don't understand this reluctance to polishing. Again, I polish all my shoes after every wearing. And they are some of the healthiest looking shoes you will see... Happy thanksgiving.