I recently visited the new Etihad Museum. Here's a glimpse of what I saw. I plan to go again because some of the spaces were not open. There's also a very good exhibition titled Emirates to the World: Postal History from 1909 to Unification. There was a no photography sign in that exhibition space, so unfortunately I don't have any images from it, but I do strongly recommend you visit.
Unfortunately, the Etihad Museum isn't updated with information about exhibition, but this link has the information about hours and entry fees.
Here are some photos I took which I shared on Twitter.