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Sara and Sarah are the exact same name minus a silent letter. They are pronounced exactly the same, and there is no difference between Sara/Sally and Sarah/Sally. If your cousin being named Sara would prevent you from using Sara, then it should, by rights, also prevent you from using Sarah.
Sally has been a nickname for Sarah for hundreds of years. Most people know this. Or rather, many people know this, but very few people care. Sally has been used on its own for centuries also, with no ill effects. Just use Sally if you like it that much.

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Sam and Jessie do work as full names and have for a very long time. Penny and Max are much more common as full names than as nicknames and most people wouldn't assume they were short for anything. However, Penny and especially Max are very common names for dogs.