Could do, though I think it will be no different than using graphite in that sitch, I think it will not take all that much before it goes all crumbly.
Natural clays are a bit like a bunch of plates sliding over each other, if you add stuff that's not all that different, like graphene is sheets of paper, thinner but flat, it might behave mostly the same still until you get quite a bit in there, whereas if you add golf balls, not so much.
Actual playdoh, I think we're talking starch molecules, so a form of carbon that's more like starch molecules would work best.
I have heard that excellent conductive paints and inks can be made with carbon black and acrylic laquer base, so if going to polymer clays then carbon black might be the thing.
Edit: By the way, I should mention I guess that carbon black, obtained as soot from a generic paraffin wax candle, is the sort of the middle of the road hodgepodge of particle shapes, like emptying out a kids shape sorter, or box of classic wooden building blocks with the rounds and arches etc. It's got flakes of graphene, nanotubes, possibly the odd small or partial buckyball, and so on.... meaning that one shape will not dominate the mixture, where one is round, another will not roll, where one is smooth, another will be snaggy, so will not in general ruin the properties of what you mix it in for reasonable amounts. Unless we get chemical instead of physical. (Something that reacts with carbon)
Unicorn herding operations are proceeding, but all the totes of hens teeth and barrels of rocking horse poop give them plenty of hiding spots.