Thanks to all.
Although I am not really looking to recover anything (in fact I don't see anything missing yet, there are "only" 12 bad sectors right now).
The only visible problem is the OS just hangs once every 3-4 days (when it touches a damaged part). I know there is something broken/lost, it's not all in the free space

... but nothing really important.
Of course, the sooner I move to the healthy drive, the better.
My main concern is to avoid re-installing and setting up everything on the new drive. I am ok if I lose the bad sectors, just don't want to get trapped for hours/days/forever trying to read them.
Thanks fzabkar, I'll look at the dd's.
I ordered some extra SATA cables and will do the cloning this weekend. Until then the bad disk is motionless.