I’ve seen some diy’s regarding painting stone countertop so you could probably do that with the ledges. You could use contact paper to cover then or you could have a wooden ledge made that sets on top of the stone one

IMO a lot mor info and pics of the stone . Are these deep window sills ? Usually those sills were done on deep ones to let you have plants . I see nothing in the space that tells me the stone is bad so more info please Take pics of the space and the window without the blinds post here in a comment DO NOT start another post,

I would worry about rhe window treatments, not the valuable stone sills.

They can be easily painted but don’t know why you would.

My bet is they stick out now bc there is nothing else to look at - once you move in, decorate, get window treatments that you like - they will recede and you will forget all about them - congrats on the move! (I would not paint stone)