Q: iPod shuffle not syncing songs

If you use automatic syncing, your shuffle can only be synced with one iTunes library at a time.  That may be why it does not sync to your new iTunes library, because it is currently still associated with your old iTunes library (on previous computer).

I have tried to restore the iPod but that is not working

Please describe how it's not working.

NOTE:  To access shuffle's Summary settings screen (if you're using current iTunes), click its device button.  The screenshot below shows upper left corner of my iTunes window, with an iPod's device button circled in red.  Click that button.