In my experience that is odd. I have experienced the exact opposite in my dogs, and I have more rice and a ton of vegetable matter. Is the Rachel Ray food also a new thing for your dogs? Because kibble is typically associated with voluminous stools. Meat and carrots certainly wouldn’t cause this problem on their own. Are you cooking the game meat? I sure hope so…introduction of a raw protein might also cause loose as well as voluminous stool, not to mention being a potential source of parasitic infection, but that’s a whole nother issue.
I went online for advice about my dogs having a touch of the runs to be blunt and it seemed many said at first try changing to a bland diet of rice and chicken then I went looking for recipes for homemade dog food and found myself here and can honestly say I will never buy dog food from the store again and sure for just sake of a little time my dogs eat healthier and have a noticeable change in just one week and I reckon cost less to feed which I would pay more anyway if I thought it was good for them.
All of the above dog food recipes will be deficient in many essential nutrients. There is no significant source of calcium or adequate sources of micronutrients such as B vitamins, vitamin D, etc. In addition to lacking calcium, the calcium:phosphorus ratio is an important consideration. Please consult your veterinarian before feeding any home diet. These are not adequate!!