My iPhone has water damage, changed the battery, worked for a while?

With water damage, you have to decontaminate the logic board before doing anything else, otherwise there will be latent issues down the road. In this case, you definitely have logic board damage.

 Not every water-damaged phone is recoverable and depending on the kind of liquid (salty, dirty, filthy), sometimes the success rate is quite low. That said, IMHO, it's always worth a try otherwise we are just piling up the e-waste.

• 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.

You should also replace the battery if it has swollen. Resist the temptation to pop it to let the gas out. A compromised Li-ion battery is a fire hazard. If the device appears to power up but behaves erratically, then use a tool like 3uTools to flash the firmware as it may be corrupted.

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.