You can do this.... but it ain't easy....
First, BACK UP EVERYTHING (files, database etc.) - TWICE. Reread that.... I've not done this myself, and if you lose what's there now because you didn't take that to heart, I'll be sorry for you but I will NOT accept responsibility!
Dump as "data only" (using phpMyAdmin) JUST the following tables from your current database: wp_categories, wp_comments, wp_post2cat, and wp_posts. Save these as text files on your hard drive.
Delete everything in your current install folder. Reupload the whole damn shootin' match and reinstall wp.
Now. Access phpMyAdmin. Empty the tables you dumped (list above). Go to the SQL tab, and on that screen, using the browse function navigate to the text files for the table dumps you earlier created, one at a time. If you get clean insertions, you should be home free.
There is the potential for errors: sometimes your phpMyAdmin dumps will have a different collation than the wp install does. If that happens, post back, I should be able to help you figure out how to edit so they work.
I REITERATE: BACK UP EVERYTHING BEFORE YOU START. I also reiterate: I haven't actually done this dance myself, though I have done similar things with other programs/databases. It should work. But that's all the hope/help I can give you.