I disagree that public history is less academic. The field applies history methodology to the dissemination of the topic to the public (plus public history theory). My undergrad program in public history required the same coursework as non-ph students. Those who decided to specialize in public history took those extra classes and went out "in the field" to gain hands-on experience. We write papers, present at conferences, etc just like traditional historians. And the bonus to being a public historian is that it is flexible and multidisciplinary--we get to attend AHA, OHA, NCPH, AAM, ALA, et al conferences based on our field of study.
OP, I'm not sure about whether switching is terribly common, but I think you're on the right track. Mostly because the museum field is more about experience than a degree. Having that history background--and a specific historical topic--while interning or volunteering during your studies will make you more competitive in the job market.
Make sure that you execute actual projects or tasks rather than do busy work or basic work that the curator leaves you alone to do (like cleaning artifacts, lol). Examples would be assisting with every step of creating, researching and mounting an exhibition. Doing community outreach. Guest speaking about the topics. Presenting your research/methods at museum conferences.
Museum professionals mean well, but the prospect of free labor from grad students who are just happy to work in a museum, can lead to them not feeling it's important to train you as a museum professional. This is your career and education, and you are their (future) colleague.