This is a good question, but not one that can have a fully satisfying answer. Personally, I want to extend the title "linguist" to everyone who studies language in a serious way. Given that, I find myself confronted with a bewildering range of approaches, and little commonality between the activities of different kinds of linguist. The range clearly includes things that could be seen as physics or physiology, others that look more like philosophy, sociology, history, geography, psychology, various flavors of biology, lots of computer science, different kinds of pure and applied mathematics, anthropology, cultural studies, and so on, even up to astrobiology and cryptography. Some of these satisfy my internal gut feeling for what sciences are like, others not so much. Even the ones that don't seem especially scientific have a tendency to fit squarely into the German term "Geisteswissenschaften". Literally, this means "sciences of the spirit", in practice it corresponds to what the Anglo-Saxons tend to call "Arts" or "Humanities". By this stage in my thinking I am inclined to reject the dichotomy implied in the question.
I think Linguistics is a good example of why we should not worry too much about whether things are sciences, and why we should worry more about whether the work we are doing will make an important difference to anything that we care about. The second question is quite challenging enough.