Becca, the number one reason a cat goes outside the litter box is stress! You haven't had her very long and it can take a while for a cat to completely settle in to a new home. The majority of cats use the litter box without any problem. However, based on the information you have provided, this new little kitty needs to be the only pet in the house. Dogs obviously stress her out and other kitties will too. A cat's sense of smell is 14 times stronger than ours, so there are odors that your cat can smell that you cannot (if you can smell the urine, imagine how it smells to your cat). There are times when a very stressed out cat needs to be rehomed where they are the only animal in order for them to settle down. Perhaps a small dog may not be an issue, but obviously, your kitty has had a lot of stress and you've got to get the stress gone. Note that Jackson Galaxy sells Spirit Essences, which are holistic remedies for various cat behaviors and there may be one that you can purchase and try and see if that helps:
http://www.spiritessences.com. There are a million topics on Google about this. As I've mentioned in a previous post, not all remedies work for all cats. Every situation is going to be different, but getting your kitty to use the litter box exclusively may require better cleaning, another litter box, and an area for her away from other pets. She needs a cat tree -- tall cat tree or climbing shelves on the wall. Let her have something that is HER territory because bengals are territorial and they love to be high up. I've never had a cat pee outside of the litter box, so I don't know if the behavior will stop overnight -- but you have to have patience and give this some time. I know how frustrating this must be for you, but outside of finding her a new, quiet, home, you're going to have to take the time to help her. Good luck.