I see three ways of tackling this:
- Get good at politics
- Enlist someone good at politics
- Start looking for a new job
If your company consistently plays the blame game, then politics are deeply embedded and there is no way around them. Are people who don't play politics blamed more often? If so, you may want to teach yourself how to get good at them.
Look at how other people determine blame for these sorts of issues, and follow their leads. Figure out who you should blame, and do it to the right people, keeping you safely out of harm's way and giving you visibility.
Playing politics isn't for everyone. Sometimes people would rather focus on fixing the problems. In that case it's good to recruit a politically-apt ally (like your supervisor or manager) who is good at the political stuff and respects your skills. You focus on solving the problem, they handle the political side.
They will be the one getting the visibility, but if there is value to what you do (actually solving the problems), it should be in their best interest to keep you around because you make them look good. This should reduce your stress as all you need to do is tell them when you find a problem, let them know how you will fix it, and trust them to handle the political details.
If the very concept of having a blame culture repulses you, it's time to take those problem-fixing skills and find somewhere else that suits you better. Brush up that resume, start applying for jobs, and do your best not to get embroiled in company politics until you are ready to leave.
To change company culture you have to be embedded in it, because companies love resisting change from people who don't buy in to the existing culture that they are invested in. It is far easier to find a company that shares similar ideas to you than to convince an entire company to change how they think about problems.