Does having just a few bad sectors actually mean my drive is failing?
In most cases yes. Some people had drives that with a few bad sectors, that lasted for years but in our experience, when a couple of bad sectors are found, more will follow. This is also supported by a study conducted by Google. They found that even one bad sectors was a very good indicator of hard drive that will eventually fail.
You can try to zero out the drive and force the drive electronics to reallocate the sectors, but that may or may not work. In the end, it is up to you whether to replace the drive or to take a chance with your data on that drive, hoping that it will last for years.