How does one go about asking someone for writing references?

You ask them. “Would you mind giving me a written recommendation?” works fine, in my experience. Never use someone as a reference without asking them in advance. Choose carefully. If you are not sure of what someone might say about you, apply for a position that you don’t really want, and figure out what your references will say about you. Fact-finding interviews are valuable at times; the goal isn’t always a new job. Information is valuable, too.

I myself have had individuals in my acquaintance call and ask former employers for employee confirmation, just to see how much information they are willing to give, as technically there is not a whole lot they should give. It has been illuminating at times.