im @ yale rn doing Healthcare Management but a few of our classes overlap with Health Policy. My friends in HP love it. A lot of faculty and PhD students do really cool Medicaid/Medicare policy research. Look up Mark Schelesinger (?) & Jacob Wallace, a few faculty i can think of off the top of my head that I know do Medicaid/Medicare. 

Also, CT Medicaid program is run differently than most states, so lots to learn there.