What can I do to fix this problem?

Nathan, unfortunately, the mistake you made was not having it properly decontaminated immediately after it's exposure to rain. You were lucky in that it worked for a brief moment but the longer water festers inside the phone, specifically on the logic board, the worse the outcome will be.

Some phones exposed to water have very minimal and localized damage but usually the water seeps under the IC's and wreaks havoc. At some point, they become very difficult to troubleshoot and often times unrepairable.

Additionally, Baseband issues can be difficult to fix permanently. Hopefully you had a chance to back up your data. You could seek out a reputable micro-solderer to try and repair or recover the data but you'll have to balance that against the inherent value of the device or data you wish to recover.