No, don't swap the records. The domain.com holds the A record and www holds the CNAME.
I just visited your site and you're running WordPress. Go to WordPress Settings > General and edit the WordPress URL and Site URL to a version with the www. This usually will create a 301 redirect for you but if it doesn't, continue reading.
Then, add these lines to the end of your .htaccess file (in root of public_html)
Code:
RewriteEngine On
RewriteCond %{HTTP_HOST} ^domain.com
RewriteRule (.*) http://www.domain.com/$1 [R=301,L]
Replace domain.com with your domain name. This will create a 301 redirect from the non-www to www. Make sure you edit the WP settings first or it will cause a redirect loop.
(The .htaccess method is the general way to choose www or non-www but settings in any CMS software need to be changed as well.)