I had the honor and privilege of visiting the National Museum of African American History and Culture last weekend, and I'm still reeling from the experience. Plan on spending at least 3-4 hours there, to get a decent exposure to the exhibits. I learned and felt so much while I was there. One of the most amazing museum experiences I've ever had in my life.
Yes, tickets are extremely hard to come by. Protip: If you are a veteran or know one, they don't have to cue up at all, and can just walk in anytime with a guest.
And as a sidenote, the museum cafe is not to be missed! Sadly, you can not visit the cafe without a ticket to the museum.