It ain't necessarily so that lower voltage will result in lower current. Probably would be in this case - but I wouldn't bet my laptop on it.
I have a device here that will pull about 2 amps from a 12v supply. Up the volts to 24, and it will pull just over 1 amp. Give it 240v and a red light comes on and it draws no amps at all.
If you leave the battery IN and connect 12v, it should be OK. Remove the battery, and I wouldn't be so sure.
Try Ebay for a 12v to 19v converter. I got one for my laptop for about £30.