How can I effectively switch fields from physics to math post-undergrad?
Sure; apply to a graduate program in math.  Most such applicants have no publications so far, so that will not hold you back.  Take the GRE.  Mention your Putnam score.  If you are in the US and are applying to a US school, you have a good chance of admission... not Harvard or Berkeley, perhaps, but at many good schools.  You may be asked to take some make-up undergraduate course work at the start; but with your Putnam score I am thinking this will easy for you.  And many math programs will let you do that after admitting you as a graduate student.  Write what you wrote here in your application.  

Another thought.  Maybe try your own university.  You have met some of the faculty already.  Email the math department, see if there is a recruitment committee for new graduate students; talk to one of them and see what they say.