This is pretty much what Foxsk8 mentioned in a comment, so credit should go to him, but these additional instructions will be useful. The WordPress plugin called TinyMCE Advanced will solve your problem.
This plugin comes with an option inside Settings > TinyMCE Advanced that will fix your disappearing <p> tags. Mark the checkbox labeled Stop removing the <p> and <br /> tags when saving and show them in the Text editor and your <p> tags will be preserved.