How do I post an image of a cover that is not yet on Amazon? I can't use the link-maker, because that's just for Amazon ASINs, right?
That is correct. To post cover AAAAAA.jpg so that it links to website URL BBBBBB, the cover must be uploaded to a server (like your own website, or Flickr, etc) so that you have an image URL for it.
Then you can use the following bbcode in your signature:
... replacing the URLs as appropriate. Let me know if that helps.