I don't really have a problem with churches involved in civic matters, even if it tends to be partisan. The Civil Rights movement started in the churches, I remember hearing MLK preaching in his home church early, preaching a sermon entitled 'Dogs in the Yard'. The idea was that if a rabid dog comes into the yard where your children play you have to take action. I heard it years ago on a vinyl LP at the library which is well before the advent of CDs.
Most white evangelicals are going to be living in rural communities who feel threatened by the liberal left. Most blacks will probably feel that the right has little regard for them or their problems so tend to veer hard left. I've done some work in doctrine and I know it can change from church to church but evangelicals are generally on the same page doctrinally. When the COGIC churches split with the Assemblies of God it had little to do with doctrine, it was right down racial lines. There was a side issue with the Apostolic (Jesus only) group but that was a doctrinal dispute between AOG and the oneness Pentecostals.
It is certainly interesting how religion can sometimes color political issues. The Catholics obviously have some political issues over abortion, I assume it effects how they vote. I'm just not entirely sure there is something in their theology that makes white evangelicals so right wing in their politics. It's got to be something cultural to have such a profound line of demarcation between white and black evangelicals in their political views.