I would not replace the battery in my G4+.
From the Motorola support website: “The Moto G Plus (4th Gen.) uses a non-removable battery. The battery should only be replaced by a Motorola-approved service facility. Any attempt to remove or replace your battery may damage the product and will void your warranty.” (emphasis added)
I have replaced my G4+, not because of a battery issue, but because of the known “ghosting” issue that phone has. Although the phone does have some age on it, I still keep it charged and use it for web surfing and streaming. I am not having the same issue you have.
I have no idea whether your issue is software related or end of useful battery life. For me the main issue is that the G4+ is prone to “ghosting” and other issues. Based on my experiences with that phone, I suggest you move on. However, if you want to continue to try to resolve the issue, the best hope appears to be to try and isolate a software issue with the help of @c1tobor or others.strong text