I definately would discourage the practice of overlaying any bad wood. You may very well get by for a while, but it's much easier to spend some extra time now to fix it correctly. This also gives you an opportunity to address any other underlaying issues that may also need attention.
Do you have the original bath components?
"One of the best lessons I've learned is that you don't worry about criticism from people you wouldn't seek advice from."
William C. Swinney