Teaching assistantships (TA-ships) should be your savior. I am on the doctoral committee of a graduate student in our department who wants to teach at liberal arts colleges post degree and therefore wants to gain as much teaching experience as possible.
She is still doing research and getting published in top conferences - just not at the scale at which other students (who want to focus primarily on research) are.
Another lesser seen but useful mode of funding is through what most schools call graduate assistantships (GAs). Here, you will be assisting in academic administration or other managerial responsibilities closely associated with research activities but not actual research itself.
I encourage you to explore those options. Departments usually have a limited number of these options.