Hi, jandghomes, & welcome. My first thought is--where is the addon domain redirecting to? If it's redirecting to the same place as your first domain, i.e., the root of your public_html folder, as an example, then, yes--the 1 site will mess up the other. I recommend you chec your CPanel & ensure the 2 sites redirect to different locations. In this sort of situation, I really like to install WordPress to its own folder, but it's not altogether necessary so long as you have the 2 domains redirected differently.
Please let us know if that helps. If it does not, perhaps providing us w/a copy of your .htaccess file will help, in order to see if there are any redirects there that might be causing the problem.