S.C.
You have a LOT of stressers in your relationship right now, not to mention some serious communication problems. My husband and I were exactly where you are right now. We watched Fireproof. We said "Wow! That's great!" and then never did anything with it. We would fight, scream, cuss, throw things, etc in front of our children. One night we had it out. I mean it was BAD. Way worse than ANYTHING we had been through before. We both admitted that we already had one foot out the door and that we were losing the love that we once had. It was terrible. I was totally heartbroken to learn how he really felt about me. He was totally heartbroken to realize how I truly felt about him.
The next day we really talked. No screaming and no arguing. We realized that we still loved each other enough to seek help. We went to counseling. It was amazing. Our counselor saved our marriage. We're in an amazing place right now. I love my husband more than I ever have before. Our intimacy is absolutely amazing (not just the sex, the cuddling, the hugs, etc.)
I can give you a few tips that our counselor gave us, but honestly, it's just better to go see a counselor. You can usually find ones that work on a sliding scale. Look for Christian Counseling places (even in you don't go to church), they usually will work with you the most.
This is an example of how a conversation SHOULD go:
You: I am feeling a little overwhelmed right now. Can you please help me with the kids?
Him: I hear that you're overwhelmed right now and would like me to help you with the kids, is that correct?
You: Yes, that is correct.
Him: Thank you for asking me, I will help you with the kids. When you're finished getting ready, could you watch the kids so I can finish?
You: I hear that after I'm finished you would like me to watch the kids.
I know it sounds totally dumb to talk like that. I was so skeptical that it would work. My husband and I made a total joke out of it, at first. After awhile, it really did help! The other thing she told us is to not go into a conversation with a "my way or the highway approach." Instead of saying "The kids need to go to bed at 8" You would say "I think the kids needs to go to bed a little earlier, can we talk about it?" If it's not a good time for him, he is allowed to say "I'm not in the place to talk about that right now, can you give me 10 minutes?" Then he or you have time to get in the right head space for a serious conversation. If at any point the conversation becomes argumentative, either one of you is allowed to say "I feel like this is no longer a productive conversation, can we please take a short break so we can calm down?" It's all about how you phrase things. Remember, it takes two to fight!
As far as the intimacy, fake it till you make it. If you don't want to hug, hug. If you don't want to kiss, kiss. It sounds weird but it really does work.
In my personal experience, me telling you this might help YOU, but it might not help HIM. He probably needs to hear it first hand from a counselor.
Talk to him about how you feel and suggest that you find a counselor. Call around and see what's available to you.
I'm sorry this is so long! I totally empathize with you. I know exactly the pain you are feeling right now!