My husband got a new E3 visa for the new employer. Do I need to apply for the new E3D?

You have a dependent visa which is not tied to an employer like your husband's. Therefore, as long has he has valid E-3 status, your status continues. You do not have to extend your status or apply for a new E3D until your visa is close to its expiration date. So if your visa is valid until December 2013, you can use it to travel until the expiration date. Your EAD is also valid until it expires. I would suggest applying for a new EAD four months before the expiration date.