R.J.
First off... you're completely right and sane to feel like you're feeling (as contradictory as that sounds)... because sleep dep makes us crazy (hallucinations, paranoia, loss of cognition, blunted or hair triggered reflexes, mood swings, the lis goes on). So you need a quick fix, Right Now.
Unfortunately there is no quick fix for your daughter's sleep, so you need a quick fix for YOURS. You need a bare minimum of 2 nights worth of sleep, and 3 would be preferable. It's time to rent a hotel room, or crash at a friend's/family member's house for a few nights. Failing either, a room/couch/somewhere in your house where you can lay down and stay there uninterrupted except for bathroom trips. THEN, after you've gotten your mind/sanity back, THEN start working out some long term solutions.
Your DH is probably feeling it as well, but your pregnancy is making things waaaaay worse, because of all of the energy that is being drained from you for growing your baby. So go get some real, honest to god sleep, get healthy, and then start working on long term solutions.
((One thing DH and I did as a long term sustainable solution was to trade weekend days/nights. One night a week I got to go to bed and sleep myself out. Ditto for him. In the beginning when I was a wreck that was sometimes 12 hours, pee, 4 more hours. Creating the night/day of absolute sleep sancity was HUGE for us, and me in particular. Even during a week from hail, I knew I had a night coming to me and I could hold on. Later we ended up loving our "days" so much, we changed schedules around (like working 4 10hr shifts, so that we had 3 day weekends. My day, His day, & Family day.))