Option 1: Create a menu for your categories and articles. Everytime you add an article to your category, you also add it to your menu. Then you use the menu module to call in that information. It'll be a menu, but it'll look like categories and articles.
Option 2: Create multiple modules, one for each category, and call them all into that space. They'll tile on top of each other, pull the relevant articles, and look as though they are one unit, even though they are several.
Option 3: Hit the Joomla Extension Directory for other article and category displaying modules and see what you find. I like Display News, but I'm not sure off the top of my head that it'll do exactly what you're looking for.
Option4: This just occurred to me. It's a little crazy, and it's more or less used to organize option 2 and make it a little more dynamic. Using Article Categories, instead of Article Category, you call the category list into your website. Make sure you ALSO call the category descriptions. Then in the description of each category, you call a module for that category that lists its articles. Now it really is one unit, but it's pulling pieces from your modules, articles and categories.
Whatever you do, make sure you have a top level menu item for your category blog, to ensure your articles take the styles you want.