That's what I thought, I just wanted to double-check. Because all of the widget titles are linked under the same HTML tags, to my knowledge it isn't possible to change one without changing all of them (you would have to alter the HTML code to do that, which WordPress.com themes don't allow). However, your blog seems to be currently using only two widgets that have titles (it looks like the rest are all "Image" widgets), so removing those titles should only affect the widgets you want to change. Insert the following command into the CSS Stylesheet Editor and then preview or save it to see if it worked:
That should remove the "Archives'" and "Categories'" title text. This command will affect any widgets you add to your sidebar, so in the future, if you want widget titles to appear you will have to remove this from the CSS Stylesheet Editor.
Adding unique images to replace individual widget titles is, likewise, impossible without altering the HTML code because all of the widget titles are linked under the same HTML tags (only a single image would show up for all of the widgets, opposed to a unique image for each widget). However, what you could do to insert your unique images is simply add two more "Image" widgets to your sidebar (your "Categories" image directly above the "Categories" widget, and your "Archives" image directly above the "Archives" widget and directly below the "Categories" widget). While these images wouldn't technically be the titles of your "Archives" and "Categories" widgets, they would at least appear to be the titles for anyone visiting your blog.
I hope this helps, and let me know if it doesn't or if you have any more questions.