It does sound like you have a short in the logic board. Likely a blown capacitor, carefully inspect for blistered or burnt components or other signs of damage. You may want to get a set of schematics and board view drawings so you can trace out the power rails as that often helps me narrow down what needs to be checked first.
Given the age of the system you will need a new battery for sure [product|IF123-054-1] so you will have that expense. So keep that in the your repair budget costs.