Don't see a backup option in iTunes for shuffle

There is no specific "backup" feature for iPods (except the iOS iPod touch). You backup is the fact that all of the songs are in your iTunes library.

This is what you should do. Create a new playlist (or more than one) in iTunes (not on the shuffle). Put all the songs you want on the shuffle on the playlist. Select the shuffle in the iTunes sidebar, under DEVICES. Over to the right, go to the Music tab. Set it to Sync music, and select ONLY the playlist you created for the shuffle.

When you sync after the Restore, your playlist is already there. You just need to select it on the Music tab, then click Apply/Sync.

Now... your shuffle only charges if you do a Restore first?

Is it flashing (blinking) amber, or solid (continuous) amber? If it's flashing amber, not solid amber, it may be working properly...

The flashing means Do Not Disconnect. Solid color (whether red, orange, or green) means OK to Disconnect.

The color indicates battery charge status. Red (very low) - Orange (not full) - Green (full or nearly full)

When flashing, it is never green, even if the battery is fully charged.

If the shuffle is set to Manually manage music (or Enable disk use), which means you drag songs to the shuffle in iTunes to load it, the shuffle's "disk" is continuously mounted when connected. Therefore, the light always flashes ("Do Not Disconnect") because you need to eject the shuffle before physically disconnecting it.

If the shuffle is set to automatically sync, then the shuffle's disk is only mounted when syncing; at other times, iTunes keeps it unmounted. Therefore, the light is solid (only flashes when syncing) and turns green when battery is fully charged, and you can disconnect the shuffle without the need to first eject it.

Immediately after a Restore, the default setting is automatic syncing. The light turns solid green when battery is fully charged. If you then set it to Manually manage music to load the shuffle with songs, the light will no longer turn green when battery is fully changed (it will continue to flash orange).

If you use the suggestion from my first reply, about setting up a playlist in iTunes with the songs you want on the shuffle, and syncing that playlist to the shuffle after the Restore (by selecting it in the shuffle's Music tab), then you will be using automatic syncing.

If you currently dragging your songs to the shuffle in iTunes after doing a Restore, then you are using Manually manage music.

If you eject the iPod and leave it connected, then the light should be solid and it should turn green when fully charged. It's possible that the battery is becoming worn out through use, and that is why it does not show completely charged, but I don't know why it does show green (for one time) after you do a Restore. If you do a Restore and get the light to "go green," set it to Manually manage music, disconnect it, and immediately reconnect it, is it still solid green or does it become solid orange (since it's no longer immediately after a Restore).

Normally, the shuffle battery should become fully charged within three hours (from empty). It may be working the same (in terms charging and available play time) whether it starts out orange or green. If you are curious, you should time it playing songs continuously at the same volume level, to see if the play time is significantly different.