What should I do about the ''HDD dying'' messages on my laptop?

The underlying gist of the message you've received is that your HDD may fail without any further warning, or data might be lost even if you think the HDD is working correctly. The large number of reallocated sectors indicates the drive can no longer be relied on to correctly read back data written to it, and the failure isn't confined to a point location (as might be the case if, for instance, it were due to a head strike from dropping the machine).

Used hard disks can be had for very modest prices, and many (if not most) laptops allow replacing the hard disk with few or no tools. The alternative is, at some point, your system failing the boot, with loss of everything you have stored on it (including the operating system you paid for, from that large software company). If you can't replace the HDD immediately, the next best solution would be "park" the computer -- shut it down correctly and store it in a safe location until you can.

Just like a car with bad brakes, even if not being able to drive it is difficult, it's less difficult than cleaning up after an accident -- and cheaper, too.