I once almost donated to the NRA several years ago. But on their main website were a few links to some anti-abortion/pro-life organizations. (And, AFAICT, it wasn't advertisements served over a third-party system.) I never went back.
I don't think the ACLU is nearly as partisan aligned or affiliated like that. But they're definitely more aligned, as a political matter, with the Democratic party. Partly for historical reasons (e.g. school busing, voting rights, criminal rights, etc.) But I totally get what you're saying.
The other part of it, though, is that we all need to learn to be a little more tolerant of political organizations and politicians taking stances that we personally disagree with. It's inevitable. To demand otherwise leads to more extremism, and leads to more fragmentation. If all we have are 10,000 smaller advocacy organizations, they're all going to drift left or right to a much greater extent. The only way to arrest that shift is to use yourself as an anchor; support the organization but then be sure to voice your opposition when they assume an inappropriate policy stance or affiliation.