I don't have a solution but I'm thinking it could be a version problem perhaps? I just started a site and am running wp 2.8 on it and just installed the plugin and it's working for me.
Here is the exact html that I pasted onto the page as recommended by the plugin.
[contact-form 1 "Contact form 1"]
It's the standard form that the plugin has created for me.
Here is the code that makes up the contact area that I am running.
<p>Your Name (required)<br />
[text* your-name] </p>
<p>Your Email (required)<br />
[email* your-email] </p>
[text your-subject] </p>
<p>Your Message<br />
[textarea your-message] </p>
I don't know if it helps or if I'm on the right path as I am new to this but are you editing the html in the tools > contact form 7 area? Like I mean your not just pasting all the code into the html area of your page or post?
I would say to edit the html how you want in the tool > contact form 7 area and then paste the code it gives for your custom form into the html area of your page or post to call in the plugin.
I hope I help a little.