Our transition with my first baby was really hard, too. I had a car, but my baby HATED it, and he screamed bloody murder the whole time. Which made it so that I only went places when I was desperate (I sometimes was!). My dh travelled at the time, and he was gone 2-3 nights a week.
What saved me??
Like PP said, just handing the baby to dh, saying, "he's yours for 2 hours. Don't wake me before then." And just letting dh figure it out. Your baby can survive for 2 hours without you. Dh loves that baby, he just isn't sure what to do with him yet. Give him time to figure it out. The baby will know that he's with someone that loves him. They'll figure it out.
I sleep in on Saturdays. Still do, and that baby turned 3 in September.
At the beginning, I'd nurse the baby, then hand him to dh, and I'd go back to sleep. Now, the babies are bigger (we have 2 now), so dh gets them up and deals with them while I sleep.
Occasionally, at night, I leave. Last night, I knocked out some shopping that was impossible to do with my kids along. I was gone for 2 hours, right at bedtime. I have no idea how he got them to bed (I usually handle bedtime). But you know what? They survived. I got some time to myself, some chores done, and even though I paid the price with an early wake time this morning (they both wanted to see me, cause I didn't put them to bed), it was worth it. Sometimes, I meet friends for dinner, or I go to the library for an hour to read magazines, or I run get milk at the store. If I need to leave, I do, as soon as dh gets home. Luckily, he comes home every night in the current job (no travelling), so I get that luxury. And, when I have a tiny nursling, I will nurse before I leave, and I don't stay gone long, but I still leave.
I really, really worked on bedtimes. 3 am would NOT cut it for me. It just wouldn't. I'd be a sleep deprived mess. There are gentle ways to get a baby to sleep earlier, by about 4 or 5 months at the most. Elizabeth Pantley's book, The No Cry Sleep Solution, has some ideas, as do other books on the importance of sleep. Sleep is important for everyone. And, even in a consensual household, if it isn't working for one member, it isn't working. Sounds like staying up until 3 am isn't working for you. You can change this. Yes, it might be hard. But, it will be worth it.
We are an early household. I aim for babies to sleep around 7pm until 5am (waking many times in there). But, I start aiming for 7pm pretty early. It is doable. And, I find that they sleep pretty well from 7-10pm or so. That gives me some "off" time. After baby bedtime, I don't do chores. I'm off. I relax. Dh has a quitting time, and so do I. I have to work all night long feeding a baby, but I certainly don't have to do laundry during that time! LOL
The other thing that helped me is to have a bare minimum of clean standards. I do have things I like, but I also let others slide, especially with a little baby. If something is bothering you, have dh hold the baby while you do it, or (even better), tell dh to do it. He'll probably step up. If he's like mine, he just isn't sure where to start, and he doesn't want to "mess up", so he just does nothing. I tell him what to do, and he does it. This isn't the ideal, egalitarian household management style, but you know what? It works for us, and it makes us happier. I see SAHM as the household manager, too. I don't do everything, but I sort of make it happen. Sometimes by telling him to do it.
Find what works for you, even if it doesn't look like other marriages.
You'll get through this!!! Babies are hard. They knock your world apart....but they are very worth it.