For better or worse, using a legal degree and manipulation of the legal system to achieve social or political goals is a wide-spread practice and one generally hailed as the most noble calling for a lawyer. This does not go unnoticed in law school and many if not most professors seem to take this approach, particularly at the more fancy schools I suspect. This sounds like possibly what you are interested in.
The other option is having the law degree as a credential/background to directly work with policy - many politicians have legal educations and I suspect are more capable for it. If you want to be a senator, congressman etc for example, legal training is very useful.
The difference here is whether you manipulate the legal or political system to get what you want. Dersh and (a minority) of other lawyers/professors would argue that manipulating the legal system to achieve your social goals is improper because it takes these political arguments out of the democratic sphere of politics and plunks them in hands of a select few elite lawyers, judges and justices. I tend to be convinced by these arguments and am very skeptical of whether so-called public interest shops are actually in the public interest. That said, public interest law is huge and generally praised as selflessly giving to society.
As a practical matter, law school all together (exclusive of opportunity costs) will run about $65k/yr and you probably won't make much $$ doing policy or public interest work. Now many schools subsidize such work (back to the do-gooder notion) to make it possible for students with heavy student loan debt to go into low paying jobs, but a look at the financials is still a good idea.
As far as schools go, Yale has a reputation as the most policy-oriented. HLS is huge and you can do lots of policy stuff or pretty strict black letter stuff.
IMHO if law school or any other school or organization changes your worldview, psyche etc then you started off on mighty shakey grounds. Law school doesn't decide who you are, you do!
As a general matter, I suggest fixing the problems you see in the world directly rather than by attempting to push your policy ideas on others. cf, if you don't like the quality of teaching in highschools, become the best highschool teacher the world has ever seen rather than another think-tank rat in Washington. For a select few popular injustices, a legal degree makes sense - perhaps you are convinced that the quality of the typical public defender is really sub-par, then you could get a JD and go become the best PD out there.
just my 2c