Why didn't they go looking for a doctor or the government?

They make it pretty clear early on that Sheila doesn't want to go to a doctor because they'll likely want to run all sorts of experiments on her and she's afraid that she'd never be allowed to leave. You can guess that trying to contact the government (by which I assume you mean something like the CDC) would result in a similar fate. And even if it wouldn't, the fear of it is what keeps them from trying to contact anyone about her condition.

I think they don't go to the government or any doctors because they would probably want to run tests on her. She also mentions this pretty early on in the first season. The government would probably keep her in quarantine or locked up because they most likely didn't know anything about it. From what we know as the audience, Sheila is the only one who is like this (besides Loki). Although killing people might not have been the best option, it was better than going to doctors or the government where they would experiment on her.