I ended up following the instructions in the article I linked to in my original post to move my Time Machine backups to a new drive. It took 28 hours to transfer 1.9 TB of backups. I then ran Disk Utility "Verify Disk" on it and it was good. So, the errors did not transfer over.
So now, I can just repartition the old drive to get rid of all the errors, which I will do after a few days of ensuring that this copy of my backups is indeed good.
I later tested the transferred backup by reformatting my iMac and restoring from the TM backup, and it worked fine (I'm on it now).