Hi @rxzero ,
If the phone stays on when the charger is connected but not when the working only on the battery, then most probably the battery is failing to hold its’ charge and needs to be replaced.
Here’s a link to the ifixit [guide|5810] guide, which may be of some help.
Replacement batteries are available online. Just search for ''Motorola SLVR L7 battery'' to get results for suppliers of the part.