I have found a way to make it work for my website so I'm interested to see if this works for you. What was happening on my site was the last three columns on every row had the following html:
<td class="thismonth right"><div class="daynum">1</div></td>
<td class="othermonth right"></td>
Don't know why is showed up like this, but because the class was "thismonth right" or "othermonth right", the html was pulling the .right CSS class from my site template's CSS file. To fix it I had to add some code to the plugin's CSS file. I've looked at both of your sites and I tested this using Firebug and it looks like it will work. James, I know you're not very code savvy, so I'm going to try and give it to you step by step.
1. Go to your admin dashboard for you site
2. Expand Plugins and click on "Editor"
3. In the top right next to "Select plugin to edit:" choose "The Events Calendar" from the drop down menu and then click "Select"
4.In the right column, click on "the-events-calendar/resources/events.css" (the second link on my page).
5. Go to the bottom of the edit box and start a new line.
6. Copy and paste the below code:
7. Click "Update File" below the edit box.
8. Go to your calendar and see how it looks.
This should supersede the CSS from your site's template (hopefully) and make it render properly. This is just a quick patch and is not a permanent solution.
Is there a way to find out why the code is being rendered with the "thismonth right" or "othermonth right" classes? I looked at the PHP and can't see right off the bat what would be causing it. Hopefully this is something that can be fixed in the next release.