Personally I don't like "Time Machine", too many bad stories about it. I use CCC because I know it works (and its free)
If "Disk Utility" is showing errors you could lose your install anyway. Bite the bullet, backup & attempt repair. Download & run "SMART Utility too.
If you installed from a "Distro" (shudder), you could run disk utility from that, or from a boot cd/retail install disk. But fsck -fy in single user mode is easy.
I did a vanilla install so OK there. But it has been quite a long time since I did the process so really hoping not to go through that again
I am in the middle of shuffling around some of my video storage disks on the machine so will try this at the end of that.