I find it quite ammusing that the authors of 508 got this so wrong... "HTML code already provides an "Alt Text" tag for graphics which can serve as a verbal descriptor for graphics"
Sorry it's a textual descriptor.

Also Katy it is worth noting that this condition does apply to 'federal' web sites and not private or other public sites.
So on a private site you can ignore these tags and be compliant.

It might be a better option to hack the css. If you make the main page contents for visual use but stop screen readers using it then add a 'hidden' css tag that is readable but hides the text content from visual display but available to the screen reader you could put your text for a screen reader in that.
It is likely that if your visual content relies on a text item referencing an image then the same content for a screen reader will need to be more verbose and descriptive so a different text content might well be a good idea.