Errant <p> tags being added to HTML

Just as I thought my problems were ending, I try to add something new! I’m trying to add some fancy tooltips using jQuery, which requires me to wrap a <span> tag around the text that I want the tool around (<a> would work as well), and then include a hidden <div> under that, that keeps the content of the tooltip (images and text).

Whenever I add the <span> and <div> tags in the “HTML” editor however, the output is incorrect, and will sometimes not show, or will have <p> tags around line breaks.

Is there a plugin that will allow me to add HTML to my posts easily?
Is this the theme’s problem, or a known WordPress issue?

Any help would be great. Thanks!!