U.S Citizen (Permanent Resident) – Washed my passport and going to Taiwan in two weeks. Need help!

Accidentally washed my U.S Passport and the profile page is still legible and still scans but wondering if I need a new one ASAP if I’m planning to travel to Taiwan for a couple days in two weeks?

I looked at the U.S Embassy website and it seems like the only option I have is to issue an emergency passport. Has anyone been issued an emergency U.S passport before and what are some points to be aware of? Will Japan’s immigration allow me to leave Japan and come back to Japan with a emergency U.S Passport?

If my washed passport is legible will the embassy accept that as proof of U.S citizenship? Or should I bring my previous U.S passport just in case?