Unfortunately the process of elimination through substitution of questionable parts with known-good ones appears to be the most likely to solve the problem.
The third part that may be contributing to this issue, is inside the computer behind the magsafe port. The magsafe board is the circuitry that controls the power inside the MacBook, and they have been known to go bad on occasion. This dc-in board kind of circuit can be the cause of symptoms, so before replacement it is good to be sure that is the most likely item. You can look into a repair guide to see what they say about replacing the magsafe circuit board behind the magsafe port. Depending on your MacBook it may be among these: http://www.ifixit.com/Device/MacBook
However older batteries with a high cycle count, or some use patterns that do not encourage long battery life, can be suspect, too. A physically old battery with a low cycle (recharge) count may also be bad.
There is this Support page with some ideas on troubleshooting batteries, says MacBook/Pro, some of this applies to older MacBook; since 'macbook' is discontinued, while MB/Pro & MB/Air continue to exist.
•Apple - Support - MacBook/Pro - Batteries & Power:
https://www.apple.com/support/macbookpro/batteries_power/
A good source of third-party power adapters for Mac portable is OWC online. They have batteries to include some newer tech stuff, and offer a lower price at times. I have never ordered from them, but had been tempted a few times. They know their mac stuff in several areas, such as RAM, SSD, etc.
Good luck & happy computing! 🙂