Update:
I was able to make the form appear how I wanted it to using Firebug and making changes to the HTML. The only problem is I have no idea where I need to implement these changes in order to make them appear on the site. Can anyone tell me where I can make changes to the HTML of a form?
Okay, to help out anyone who wants a similar effect this is what I ended up doing, and it works. First create the form normally using Custom Contact Forms. Drop the form on a page using the shortcode, update it, open the page and view the source (right click > View Page Source). Copy everything from the source in between the tags <form> </form> and paste it into the HTML tab on that page. Your form should now appear exactly the same as it did using the shortcode and you can now do whatever you want to the HTML.
Note: I had to download a plugin called Raw HTML by Janis Elsts to stop WordPress from generating paragraph tags and line breaks. Once you download the plugin put the form HTML in between <!–raw–> <!–/raw–> tags and you should be good to go.
