wow. Marble slabs are used in candy making because of their ability to stay cold or absorb heat, so you have sills with unique properties. Can you show a pic?

eta: For the coldest months, I'd probably just find a way to cover them with a thick fabric (e.g., poly fleece) to act as insulation.

This is why old houses have heavy insulated drapes over all the windows. Its also why the spray foam guys stay busy drilling holes and spraying in the stud cavities. But only after all the knob and tube wiring is gone and the house rewired, plus the rusted galvanized and cast iron is gone, and the house is replumbed.

And the windows are rebuilt to be more energy efficient. TOH has s great article on creating better energy efficiency in original windows. it involves rebuilding the sashes and adding storm windows.

If you cover them, unless airtight, you are just going to have condensation.