What to do about the MIL? Do I stay with her son and put up with her nonsense or give it all up?

I've been dating my boyfriend for over 2 years now, we just recently had a baby. We couldn't be any happier. We have a really good relationship, but his mother is getting in the way. I can't do anything right , even simple stories I tell her when she is here get twisted and turned around. She has a problem whenever I say something my mother did. She comes out with out right lies, and sends emails to my boyfriend in all caps about how bad I am and how I'm this and that. She was mad because we made her wash her hands after she smoked before she could feed/hold the baby... who is not even 2 weeks old! She's mad because she isn't allowed to smoke in our apartment, so she has to go outside. Well my mom and anyone else does the same thing, and even if she didn't smoke she would still need to wash her hands because of germs! I'm a first time mom, I worry about everything, and trust no one completely with our baby. That's normal, at least that's what other moms tell me. I'm really sick of her sending my boyfriend nasty emails about me, and saying negative things behind my back , but then tries to be nice to my face, because she wants to see the baby. Both sets of parents live an hour away, my parents have been down here more times than I can count, but we always have to go to her house and if and when we don't, well than it's my fault. Everything always turns in to it being my fault. For the last 7 weeks of my pregnancy I was in Pre term labor, I was put on bed rest, and had contractions all day long. She never called me to see how I was, she never came down to take me to my three doctor appointments a week, she never came down to help me around the house, and even now I had a c-section, and she never even asked me or my boyfriend how I was feeling. It's a major surgery and I was in a lot of pain for a couple days. My mom came down and stayed with us for a couple days and she helped clean because I'm not supposed to. But yet she claims to be all high and mighty never doing anything wrong, but really could care less about anyone but herself. Also, I should say that she only sends these emails at night when she has been drinking all day... not that that's an excuse but it's fact. I do have to say that my boyfriend just deletes the emails and never brings the subject matter up to her again, he just ignores it, says he's sorry and that it's stupid what she does, but he can't do anything about it. I almost feel like I don't want to continue fighting with her and having her be so nasty to me, for the rest of the time that my boyfriend and I are together. I shouldn't have to put up with someone like that, and I love my boyfriend, but if she were a friend treating me like this, than I obviously would stop myself from associating with her, should I do the same to his mother? Especially with the new baby?