Magsafe and Battery Charger Board serial numbers?

Mackbook 4.1 A1181 early 2008 core 2 duo 2.1 ghz 4gb ram

Hello, I have this laptop and it doesn't seem to work with the battery but it works when plugged in with the charger.

I'd like to replace both the Magsafe and the battery charger board and see if that fixes it (as it wouldn't be so expensive to do both)

But I don't know exactly what part number is compatible with my macbook.

I have these ones booked marked:

Magsafe, 820-1966-A

Battery charger board, 820-2029-A

Is this correct?

Also I've heard that this can also be caused by using a non-OEM charger, and non-OEM battery.

I use both but I have no way of testing it with the original apple products.

I will try if replacing these parts doesn't work and just hope it's not the logic board.

I'd appreciate very much any help. I'm clueless.