I know a LOT of former mainline Protestant church goers who've switched churches (sometimes even within the same brand) for exactly that reason.
Generally speaking you should be registered at a particular parish-—like being registered to vote.
However, Catholics are free to attend Mass anywhere they want. And we do attend Mass at various different churches in the area on Saturday night depending on where we are as Mass approaches. On Saturday night we have one church that starts Mass at 5:00 PM, one that starts at 5:30 PM, and another that starts at 6:00PM. So we have a lot of choices and a lot depends on where we are and what we’re doing.
Fortunately most of the services we attend are free of political discourse and concentrate on things churches should be focused on.
However, when go to the main Cathedral in downtown, where I am registered parishioner and volunteer for Sunday morning Mass, we are often are subjected to the Archbishop’s exhortations on immigration reform (amnesty) and the Cathedral is also helping the state of CA implement its new law allowing illegal aliens to obtain drivers licenses