I would think that a utility knife would remove the caulk easily, and a
grout removal tool is very cheap and easy to use and will throw much
less debris around than the dremel.

Some food for thought - if the tiles aren't loose and it's just a grout
problem, have you considered just covering it with a tub surround?
They're pretty easy to install, look good if you do a decent job on it,
and can come with all sorts of shelves and hanger bars molded in.