Chosen Solution

If my iPod doesn’t hold charge is it the battery or the logic board

Battery. If it boots with a 10W adapter, then the battery needs replacing. Replacing your battery requires a solder and is very hard, similar to an iPhone 2G. iPod Touch 3rd Generation Battery Replacement

Most likely it is the battery that needs to be replaced. If the ipod were damaged in some way then there could be a possibility that the logic board is damaged but the only way to tell would be to replace the battery.