The bottoms of my feet often get that way. When I (rarely) get a pedicure, I refuse to allow them to aggressively file my feet because years ago a pedicurist actually made my foot bleed.
So I use a gentle, homemade exfoliant of granulated sugar and oil, sometimes with a little essential oil. Super cheap and easy to make, and I know exactly what is going on my body.
For oil I use baby oil, grapeseed oil or light olive oil.
For scent I use peppermint (give feet a nice tingle), lavender (relaxing) or sometimes ecalyptus or rose. On little-boy feet that have ground in barefoot grime, I use tea tree oil for the scent, no more smelly feet for several days.
It also works on hands. After wards, just rinse, pat dry & moisturize.