ListMail sends the HTML code only.
Imagine that you are an e-mail client (Outlook) that opens an e-mail. It reads the code and starts looking for images. These images need to be somewhere like on your Web site. I would not host these images elsewhere.
I design the e-mail in Dreamweaver and upload the page (e-mail) to my Web server. As Dean mentioned, the HTML code must use absolute URLs to be able to find the images.
You can not embed the images in ListMail, then need to live on your server and the e-mail client goes out and grabs them from the server when the user opens the e-mail.
You stated "I copy and pasted the html code into the emails and sent them". This may or may not (as you found out) work. The images from that page were trying to grab images locally, and they were not around. Hence you need to change all image references to be absolute.
You also noted that "They came out horrible". An other issue may be that the html code you cut and pasted may be referring to a style sheet, which could very well distroy the look and feel of the font and tables of the page.
My suggestion would be ...
Build the HTML in Dreamweaver or an HTML editor.
Upload that page and all images to your Web server somewhere.
Ensure that the links to the images are absolute references -
http://domain.com/images/marketing/header1.gif as an example
Then copy and paste the code into the HTML box of ListMail.
Concerning the text version, you need to just put straight text in there and nothing else. For links, you need to put in the complete URL -
http://domain.com/contact/contactme.asp or whatever.
Does that help?