Is my motherboard damaged?

You'll want to decontaminate the logic board. If you want to try this yourself:

• Open your phone and remove the logic board (follow this guide)
• Inspect the logic board, especially around the connectors and look for corrosion.
• Inspect both sides of the board. Unfortunately, most of the board is covered in shields. That's usually where the damage is occurring.
• Put your board in a container with >90% isopropyl alcohol and let it sit for a while.
• Use a soft brush, like a toothbrush and lightly brush away any corrosion you see.
• Rinse in alcohol and repeat.
• Let it air dry for a day.
• Re-assemble and hope for the best.

A professional repair shop that does water damage repair may be able to recover your phone or the data because they have access to pro-level ultrasonic baths and specialized cleaners as well as the skills to troubleshoot your board. Many shops have a no fix/no fee policy so you don't have to spend money to find out if the phone is fixable or not.