Cat litter box problems?

Lady adopted a shelter cat, at the shelter the cat used the box fine. Now at home we are elminating next to the box. The box is uncovered, don't know how often cleaned but I recomended daily changes to see if its just a matter of cleanliness. I Recomended Cat Attract Litter. Then the cat has been checked by a vet, no UTI but does have a severly ifected tooth not sure if that is affecting this at all. What else should I recomend?