Posts are correct in that teaching experience is nominally, and negligently, unnesessary for Ed policy work. I finished my corps commitment and am currently teaching abroad, en route to grad school in edu next school year.
Like you, I toggled between grad school right after or TFA. Part of my SoP actually details my thought process. I was super anxious about grad school, but I had already finished my cert to teach secondary social studies as an added major during undergrad. I figured TFA, with all of its cultural incompetency and faults, would offer me unmatched insight into the world of edu I wanted to one day help improve--- so I went with TFA and don't regret it.
Honestly I think it's best to have some classroom experience before braving Ed policy. Too many policy issues exist now that could have been mitigated or completely avoided provided the people creating policy had some actual experience. Personally I find it incredibly problematic that people are allowed to advise education lacking experience. That's unacceptable in most other professions that society considers important, teaching shouldn't be any different.