I2010 Macpro with corrupted OS

Your old one is too old to transfer over the MacOS. So you use Migration Assistant (or Setup Assistant if you are ready at first run) to transfer only your files, not MacOS.

You could also use Recovery Mode to repair your Disk, and/or to install a new macOS, right over the old one. By design, this does not touch any of your files, only system directories.

The one thing you may not be able to do on a 2010 model is ERASE the entire disk, unless you boot from the Recovery partition on your Time machine backup drive.

You need a Backup:

If you do not have a recent local, disk-based backup, your computer is like a ticking Time bomb. You are only one disk failure away from losing EVERYTHING! Drives do not last forever. It is not a question of IF it will fail, only WHEN it will fail.

If you are using another direct-to-disk backup method that you prefer, and you currently have a recent disk-based backup, that is great. If not, you should consider using Built-in Time Machine. Take steps to acquire an external drive as soon as possible. If you buy one, a drive 2 to 3 times or larger than your boot drive is preferable for long term trouble-free operation. Do not pay extra for a drive that is fast.  (You can get by for a while with a "found" smaller drive if necessary, but it will eventually become annoying).

Attach your external drive and use

System preferences > Time machine ...

... to turn on Time Machine and specify what drive to store your Backups on.  It may ask to initialize the new drive, and that is as expected.