Thanks for providing a link! When I look at this page I see:
<a href="home.php?cat=737" class="arrow" onclick="javascript
: return switchSubcatLayer(this);"></a>
For the arrow link, and:
<a href="home.php?cat=737" class="icon-link sub-link">Shop By Room</a>
For the text link.
These would need to be the same. But I assume you only want this behavior on the first level of categories, otherwise you could not navigate to the actual pages.
Accomplishing this might be tricky. The way the flyout menus work is by looping through some templates, we would need to change the template code to check that it is the first level and insert the link differently if so. Look at
/common_files/modules/Flyout_Menus/Icons/fancy_subcategories.tpl
Another way to get the behavior you want is to change the type of flyout menu that you are using, look at General Settings -> Modules -> Flyout Menus and try changing the type to "Flyout Expanding" rather than Tree-like.