I wanted to update this. I've not been playing the Sims much lately, but I finished the vacation in China, went home, and tried again. The adventure board is still showing the exact same adventures. Jade, Nectar, Emperor's Tomb, Writings of Qin Shan Hu. It doesn't matter if I complete them, it doesn't matter if I get the other Sims in my party to accept the adventures and not complete them.
At this point, I think I'm going to have to accept that I'm never going to get Pangu's Axe with this household. I'm curious to know if I'm the only person having this problem though?