It does sound like a power issue. I would try replacing the easiest thing first, which is the power button board that sits just behind the power button. Sellers on ebay sell them for under $10.00. They are not terribly hard to install. Remove the battery and unplug the power. Undo the screws from the bottom of the laptop. You just have to remove the palm rest, the speaker/keyboard bezel, the keyboard, and the screen bezel. Loosen up the right side hinge and you will see it. One screw and a tab hold it in place. The wires are extremely fragile and break away from the power button board really easily. Bless Dell's pea-pickin' heart. They designed the power button to be right where the hinge is, where the wire gets the most movement.
When you replace it, you want to make sure that the wire is taught from the connector end and has lots of slack at the hinge end.