I'm not sure how putting in a fresh battery would do anything more than melt wires quicker rather than fix something.
You probably have a poor ground on the engine if the problem arises when you try to start the car.
If it happened/still happens as soon as you turned the ignition on, then probably not. Usually it is when people try to crank the engine with a bad ground, which is when the other earth points such as every black ground wire or the choke cables etc. carry currents they were never designed for and heat up in the process. If every other electrical system still works then you may have dodged a bullet although much of your harness may be on borrowed time. If everything does work fine then before cranking the car connect a heavy jumper cable from the negative terminal of the battery direct to a chunk of the engine somewhere, like a head but or lifting bracket. If it now cranks fine with no smoking wires, install/repair a good earth strap.