Hello Everyone, thank you for the responses.
Jan, I will try out this trick today, that sounds like exactly what I needed.
Brenden, I'm using unique class names and making my styling as specific as I can to avoid bleed through. The problem I'm having is that all of the unique classes I assign to my web parts and containers also appear in the design and page tabs, so their styling appears there too.
I suppose the problem could be related to the way I'm building my site? Am I not supposed to use the "Web Part Container" fields to add classes to web parts?
Actually, would it be possible for me to use Jan's trick within my Web Part Containers? I am currently using this:
<div class="{%ContainerCSSClass%}"> "web part goes here"</div>
but maybe I could change it to something like:
<div {%PortalContext.ViewMode=="LiveSite" ? class="ContainerCSSClass" : ""|(identity)GlobalAdministrator%}> "web part goes here"</div>
Would that work?