How do I show a rendered image in a new viewport without the verticies being highlighted? Sorry it's a confusing question. Here's an image of what I'm talking about.

 

I would really like to see my model rendered without all the black outlining everything as I work on it.. I never had this issue before back in March when I last worked with blender. 

Edit: The material is shadeless and the rendered mode is selected for the view. I realize you can't see that in the screenshot.
I'm not sure if it is the ideal fix, but if you just want to not see the vertices in the view on the left (which show up because toggling into Edit Mode on that object is affecting its mode in both views), I think turning off Overlays in the left-hand-side view will give you what you want. I've indicated where the relevant button is below, and give an example of what the result looks like:



As alternative, there may also be a way to only switch mode on an object in one viewport window but not in another, but that I am not sure about.