Does this mean my battery is going?

I would say yes - there is a problem with the battery - but don't forget - the battery rated life is way shorter than in newer models

on the iPod wiki page is a list where you can compare them - the times are for new batteries - so a used one can be slightly off . your model should work around 8h with a new battery (that means continuous play - not every 2 seconds switching the song ;-) )

as i mentioned in one of your other questions - i would sell it and get a different one from ebay - if you need looooooong playtimes - a nano 3g works great - have one that playes 25.5h in shuffle mode, my mini plays around 5.5h - but thats ok - the plus side of the mini is the compatibility - if the 4gb microdrive dies - i can put a cheap compact flash card in it, my nano 1g works for 6.5h (times always in shuffle mode)