My MacBook charger is supposed to pass 60 watts power, but it's passing only around 12 watts. Also, the green LED just blinks randomly.
Should I get a new charger or something can be done to existing one?
Green LED just blinks randomly.
This is most likely an issue with the DC-In or your logic board.
This is an error condition caused by a fault that is detected between your MagSafe and the power circuitry. There's a small possibility that the issue is your charger, but it's highly unlikely. The easiest way to test is to take your MacBook to Apple and hook up one of their chargers - if the blinking persists, it's your DC-in board or your logic board.
If it's your DC-in board, the procedure to fix is is pretty easy and is a DIY. Depending on the model of your Mac, the DC-in board may be integrated into the logic board itself; which means you will have to take it in for service.
If happens to be your charger, then you will need to replace it with another - there are no user serviceable components. Be sure to get a genuine adapter and not a knock off. I've seen way too many times the result of being "penny wise and pound foolish."
Is your MacBook charging up properly when you are using it with the charger plugged in?
It might only be drawing 12W of power because your batteries are fully charged and the 12W is only used for your current operation.
Having a charger capable of giving 60W does not mean your MacBook will always require 60W from it.
For example, my MacBook Pro is currently drawing 20.31 V from the power adapter right now.