I'd definitely vote to replace the battery connector before the DC-in. Battery connectors go bad all the time on A1181s, whereas DC-ins very rarely go bad. Especially if the light on the DC-in is solid green, I would not suspect it. Battery connectors are often not pushed all the way into their sockets (or they become dislodged over time, especially the models that do not have the metal piece that holds them down), so it's worth giving it a good push downward before buying a new one.
But definitely -- if you have a solid green light, look at the battery connector before the DC-in.
And also, don't trust a battery that you have not personally seen charging with your own two eyes. It's not good enough that it's a new battery and "should" work. Borrow a friend's that you have seen working.