Water damage on cardboard pieces
Here's what I did, and they don't look bad.


Used a towel to pat dry the tiles.
Used my fingers to press out the wrinkles and squish the expanding cardboard back together.
Laid them down between two towels, put a textbook on top, and pressed down to get some more water out.
Pressed out the wrinkles and squish cardboard together again.
Laid them down between two paper towels (paper towels are thinner and I wanted to make sure they stayed flat) and laid several textbooks on top.
After several hours, squished with fingers again.
Dry paper towels and textbooks again.
The next day, I pulled them out and laid them on a dry towel.


At this point they're holding together really well and are only marginally thicker than the undamaged pieces.



The damaged piece on the left is only a bit thicker than the dry one on the right.