My Sony Cybersshot DSC-T70 digital camera keeps on vibrating. If I replace the battery, would it solve the problem

No, it won't. It's a image stabilizer problem, a well know problem with many Sony cameras. The stabilizer or even the whole lens unit has to be replaced. I suggest you contact your local Sony service center and see how much they ask for the repair. The thing is that there are some used units being sold around on eBay etc., but replacing your faulty unit with an used unit, it can brake down again sooner or later. Sony now supplies improved stabilizers for these cameras, but only Sony service centers can order them.
Tom