Update on 'water damaged' visa page in passport.
I was advised to get a new passport. With the new passport in hand, I was able to get my Thai visa transfered to new passport.
I was worrying that I had to go back to Immgration office in Phuket, where visas were issued. No need to worry ... if your visas issued in Phuket, you get visa stamps transfered on same day. If visas issued in a different cities, transfer done next day.
The lesson: invest in a ziploc bag when cycling during the moonsoon.
TLDR : Answer Summary
After experiencing water damage to your passport, it's advisable to get a new passport. With the new passport, you can transfer your Thai visa. If your visa was issued in Phuket, the transfer is done on the same day, while if issued in another city, it will take until the next day. A practical tip mentioned is to use a ziploc bag to protect your passport during the monsoon season.