Those dark areas are definitely from skin contact. I have similar marks on the desk I'm setting at right now. It is red oak with 4 coats of water based varathane.
If the oils and acids have penetrated the finish and gotten into the wood you might have trouble getting it out. In that case I would take it outside and use some oxalic acid to attempt to remove and brighten it....use it on the whole table. If you have trouble finding the oxalic acid a deck brightener from the hardware store or boat shops have teak cleaner in which you sprinkle on the wet surface and light scrub away the dark color. Rinse thoroughly. Repeat as needed until you get the results you need.
I would doubt that using an oil finish like Odies oil will prevent this but it might make it easier to refinish the next time. I think the skin oil would just join the Odie's oil and wax mix and become part or the wood.
An alternative hard finish might be a lacquer which may resist the skin oils better.
Last approach make your husband wear long sleeve at the table....LOL