ETA: Probably way, way too late to the party, but in case it helps in the future or helps another poster:
If you use cream of tartar by itself, you will get some dense, sour-tasting dumplings. As stated above, the cream of tartar needs a base to react with to form the bubbles that leaven the dough (remember the vinegar & baking soda experiments of elementary school science class? Same idea, with cream of tartar, aka tartaric acid, as the vinegar. Feel free to dip a finger into the cream of tartar and taste what it’s like when it’s not neutralized. Sometimes, there are baking dishes where the cream of tartar is purposely not completely reacted with baking soda, like in snickerdoodles. Just leaving a bit of the tartar unreacted gives them their characteristic tang [which most commercial versions lack. A snickerdoodle is not just a cinnamon sugar cookie!]).
Anyhow, where was I? Better to leave it out and make a smaller dumpling that does not require leavening. It will obviously be a different, chewier type of dumpling, but you can make something like spaetzle or potato dumplings and it will go well with the dish. Or make egg noodles. I know, it’s not chicken-and-dumplings at this point, but it gives you some options.