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Do you see an arrow or the progress dial? All of those things (except for the time) would tend to indicate that the hard drive is going bad, especially if you see a spinning progress dial rather than a pointer. I certainly don't think changing the PRAM battery will help with any of those things.
Do you have a good battery? While there is a PRAM battery, the main battery keeps the PRAM battery charged. So as long as you have a good main battery, the computer should be able to keep time.