I feel for you, I'm in a similar position. I have a baby who is almost four months, and it's amazing how much easier it is now than it was when he was 8 weeks. A little more sleep goes a long way toward feeling better about things.
I wouldn't worry too much at this point about your husband's relationship with your kids. He will have an opportunity to develop that later, when they are older, and perhaps he will have less pressure then. Work pressure tends to be cyclical and often gets easier as people are at their jobs longer. My husband didn't really connect with our daughter until she was 3 or 4, and now they are VERY close. My middle son is 2 and they are just barely starting to connect, but having seen the process with my daughter, I'm not worried, I know it will happen.
As for your relationship, I HIGHLY recommend a date night, as often as you can. Maybe once a week (that's what we have) or once every two weeks. If the budget is tight, just go for coffee, or a drink, or a walk. If you can afford it, treat yourselves to a meal. It doesn't have to be long - we usually go after the kids go to bed (around 7 or 8) and are only out for 1.5 hours, but it does WONDERS. We really reconnect in a way that we can't when we're home and distracted by everything we have to do. I'm sure you can find a babysitter who can come regularly on a week night.
Mostly, be patient, this is the hardest phase in your life. It will pass.