I do want to include the Alt Text for all of the pictures that I include on both my posts and pages, but I'm finding that their inclusion within the image "box" really throws off my page flow, especially when using floats.

Is there a way to have the Alt Text attached as usual, but simply not be visible under the images?

not sure what you mean, can you point to an example?

Sure, just go to my home page: TheCharlotteDentist.com. About 2/3 of the way down the page, there are 2 images with captions under them. That text is actually what I wrote in the line for Alt Text when I uploaded the images. That text is what I want to be there as Alt Text, but I don't want it to actually show up under the images.

Did you try changing the styling of images via Appearance>Atahualpa Theme Options>Various Content Items>Style Images ?

Perhaps under the box Captions Text: Style (I haven't fiddled with it myself, so can't say if it will for sure work -- but that is where I would start)

if you want to have that text show up when you put your mouse on the image, put it in the TITLE not the alt text. This is not something that Atahualpa controls, it is WordPress.

That raises the question of whether or not the image "title" carries the same SEO weight as text defined as the image "alt-text"...