The only time the link is broken is if I change the date after I publish, right?
Yep, that's right. The reason why it breaks is that once the post is published, your followers are sent a link to your post — and that link includes the date the post was published. If you change the date after that, your post will have a new link and the old one won't work anymore.
I have a post that is scheduled for January 3. Since it has not been published, if I changed the date to today it would create a new link?
If you change the date on a scheduled post, it will change the date in the link to the new date. But it won't break anything, since the post hasn't been published yet. :)
Do I have to change the date before I publish something, or when I hit publish does it do it automatically?
If the Publish box in your post editor says, "Publish immediately," it will automatically give that post today's date when you hit the Publish button.
Please let me know if I can provide any further help with this. :)