I haven't read others responses, so I might be repeating them.
I can TOTALLY relate with you though. I have a 2 yr old and a 5 wk old, so things have been crazy at our house too. It seems impossible trying to balance enough time for each child, especially when a newborn needs more care and takes so much time to do it. I would say just try to spend enough time/give enough attention to her during the day, so she isn't feeling shafted then also make sure YOU are the one to read her a story, sing her a song, and cuddle on the couch a little bit before bed.
I agree with your husband in a way about not going to her when she's throwing fits and such because then it teaches her that she can do that and get her way. YES her life has been turned upside down and she may feel a little resentment, but that does NOT mean she can throw fits and misbehave in that way to get her way. She still needs structure and rules. If she truly needs comfort and needs some loving, daddy can and should be the one to do that. You need to remember that you can't do everything. Even if she seems like she just wants you, daddy is going to have to do. Life is different right now and she can have you at other times, but daddy can be the one to comfort her at night. So she will just need to learn that she gets daddy or no one at that time. It'll take some adjustment for her, but if you're consistent, she'll get the idea. (and it's not "i can't go upstairs", its "daddy is the one to help you at nighttime" and I would try not to say "because I need to be with the baby" because then I think they learn to resent the baby because they are the reason mommy's not coming.)
You need to focus on the baby at night and try to get the sleep that you can.
Our son was doing fine when the baby first came home, but then after a week or so bedtime turned into a horrible 2 hour ordeal of screaming and throwing fits, but daddy was the one to help out. He is better now. But has given up his nap, so he has meltdowns in the evenings. So that is still something we're trying to work through.
Do what you feel is right, but you need to remember you can't do everything and she can't always have you! If you give her love and reassurance during the day, she'll be fine. Remember the "terrible twos" are the times of throwing fits and power struggles.
Good luck.