We've been through way worse. The counseling we received focused on one thing, Love is an action. I used to feel the way you feel, until I stopped expecting the marriage to make me happy and realized it was up to me and him to make a happy marriage. It's like a living breathing member of the family, and if it's ignored and not nurtured, it will die.
It sounds like you have a really good guy and you are now out of the Honeymoon stage. With little kids, it's hard to remember to focus on the marriage. I know you said he's not social, but you need to spend time alone with him regularly and work on the marriage. Find something you both like to do, or pick up a new hobby, and do it once a week. You don't have to leave the house. My husband and I would often have date nights in the living room after the kids were in bed.
We've had really hard times that almost broke us up, stuff that would have broken up many marriages here (at least, that's if people really mean what they say about certain things being "deal breakers"). But we have four children, and that was enough to force us to plow forward.
Marriage is work. It's sad you don't have more support from your family. The part about not feeling attracted to him, well, we all go through that. Keep talking to him about it, it sounds like he is at least willing to listen to your concerns and work on being a better husband and father. And it sounds like, although you are frustrated and confused, you are willing to work on this to achieve a good outcome. Sometimes in a marriage, you just have to wait out the bad times. Nothing you've said here makes me think you guys can't make it work. We all change, we all grow, we all adapt.
And you are burnt out. I was an assistant teacher for two years when my kids were young. That doesn't even compare to the work you do as a nurse, but I bring it up because I remember how awful things got for us during that time. It was a time I felt totally uncared for, and totally burnt out at the end of the day, and I was just teaching! You take care of sick people! When do you take care of you? When do you take care of your husband?