If it's fully functional, use the usual method to copy them to your PC (sorry, I don't know what that method is, but there should be one). Then the "broken" part of your story isn't relevant.
If it's only half functional, and the usual method doesn't work anymore, it depends on the exact defect, I think (or, said otherwise: what half exactly is functional).
Besides, this might prove the necessity of backup. With phones, you never know if you lose them, or drop them or when they fail. So it makes sense to backup everything you don't want to lose to (for example) your PC regularly. Between the lines of your question I read you didn't do that. That means taking a certain risk, as you might experience now.
Is there ANYONE ANYWHERE who can tell me how to do this. I cannot lose all of my messages when I get a new phone (mine is broken but functional) I simply want all of my text messages to go from my old , broken Palm Centro to a brand new Palm Centro (or onto my computer, for that matter). PLEASE HELP!!!! Thanks!