Are you sure that the motherboard socket is really shorted, and the reading was not the result of capacitors charging?
Are you aware that many (but not all) Dell laptops use a three-connector power socket? If not, read on...
Look closely at your power socket on the MB, and the power plug on the adapter. The three connector types have the power supplied by the inner and outer sleeves of the plug, while the centre pin is actually a control pin, not a power pin. You should be able to see the contacts on the inside of the socket that would mate with the two sides of the sleeve on the plug.
So, if you have that type of connector, and have measured conductivity to the centre pin, you are not testing what you think you are.
The centre pin on these adapters is used by the laptop to make sure that the correct PSU is plugged in and, if not, to not allow proper operation. Really, it is to make you buy a proper Dell PSU instead of using a cheap replacement if the PSU fails.
Check out your connector and let us know if it is this three-connector type.