Can I create an autoresponder that sends the new contact the latest email newsletter that is associated with the contact list that they've just signed up for?

You can set up an autoresponder to contain the latest newsletter and then set that up to be sent immediately after sign up.

From here what you can do is send your newsletter out when it is due to be sent and then update the autoresponder to be this newsletter. This way all of your existing contacts will receive the email when you send the campaign and all your new contacts will receive the newsletter by way of autoresponder.