The Categories Widget is meant to display all the categories you create when writing posts, so there's no point in using it twice: you only get the same list again, as you've seen. And it only displays categories, of course, not post titles.
Perhaps you can approximate what you want by adding the Page Widget, creating multiple pages, giving them the names you wish and putting links to the relevant posts in each one. But again only the page names will show in the sidebar; the post titles will show in the main area when you click on a page name.
Note that both categories and pages can have a hierarchy (by assigning some of them to "parents").
If what you're looking for is ordered sub-lists of post titles, another option might be to switch to one of the very few themes that display titles instead of posts when you click on a category.
If you really want the titles of the posts to appear in the sidebar, you can add multiple Text Widgets, give them the names you wish, and put the appropriate post links in them.
Please let me know if I have understood your question correctly - or if you need further details on any of the above.