> women's' suffrage happened when it did so they could vote for prohibition
Woman's suffrage happened after the prohibition. The prohibition was the 18th amendment (taking effect on January 16, 1920), and woman's suffrage was the 19th amendment (ratified on August 18, 1920).
You're right about the timeline of amendments, but keep in mind that in western states (as well as in Michigan and New York) women had complete voting rights before 1920. In most other states they had partial voting rights. And since divorce was uncommon at the time, voting "dry" was a big priority for women voters.
Woman's suffrage happened after the prohibition. The prohibition was the 18th amendment (taking effect on January 16, 1920), and woman's suffrage was the 19th amendment (ratified on August 18, 1920).