Sasha, it sounds like you may have figured out the answer. Oozing is a useful word because it describes a particular way that a liquid can escape from a container (whether that container is an animal’s or a person’s body, or some inanimate object). It describes the process of escaping SLOWLY and usually maintaining contact with the body/object as more and more of it slowly leaks out. For example: “Blood was oozing out of his neck” is very different from “blood was shooting/spurting/spraying out of his neck.” The latter emphasizes that the blood was coming out fast and traveling quite far from his body.