DreamHost has all the apps that DHA has. If you pay for hosting here, be sure to use the Advanced One-Click installs. That will give you FTP access to your site and you'll be able to upload your own themes and plugins.
I know that if you pay for hosting and use the Easy one-click, like on DHA, you can retain your database. In your case, you'll have to contact Support to get them to migrate your data to your paid account.