Unless you start dressing in bedsheets and setting fire to crosses, you're not too white.
I like listening to Classical music too, Beethoven is also a favourite of mine, but I never consider this to have any bearing on my skin colour or ethnicity. I also like to listen to Rap and Hip-Hop, but I never feel myself becoming blacker as I do so.
Besides, even if your appreciation of Classical composers means that you are as "White as white can be", why is this a problem?
I think the trope that being white is uncool and that because something is associated with being black it's automatically cool or better, is quite racist... and I think it's sad that some people feel the need to parrot (and inadvertently lampoon) black culture in order to be 'cool', rather than just being comfortable with what they are and liking music for strictly musical reasons.