It depends on the specific model you have, but generally speaking I'd say just get a new phone. A lot of phones sold in the past 5-10 years have gotten rid of the user-serviceable battery. It's technically possible to replace the battery in these contained devices, but it's not an easy task, and there's no guarantee some hardware integrity system isn't going brick the phone when you boot it up the first time. Buying a new or used replacement device is probably the better option. 

 be sure to wipe your device before taking to a service business to replace the battery 

 Some devices have a "valet" mode. You activate it before bringing it in for service and it protects your data from being accessed.  

 yes, new Androids have it.  I don't trust it yet 

 Wiping your device is still the primary method of protecting your data recommended by manufacturers, but the valet mode, or "repair mode" is a good second option to consider. Here's a good explanation of how repair mode works on Android:
https://www.pocket-lint.com/what-is-google-pixel-repair-mode/