I try to sync but always get the 'other library' question 'Do you want to erase this iPad?'
I do the right click Back Up and the other one about authorize this computer but then after sync half my apps are gone and I have to restore from back up and then it asks again if I want to erase this iPad!

Maybe I need to erase the iPad and start all over again?

Here is Apple's recommendations on how to transfer all your stuff to a new iPhone. With a few modifications it works very well for moving your iPad to a new computer. Wish I could find one as well written for the actual task.

How to back up your data and set up as a new device

Start by going into iTunes Preferences (before connecting the iPad) and turning off auto-sync. It is under the Devices tab. This will prevent any unwanted syncing before you are ready.

Do only the parts that apply to your information/situation (i.e.; if you are syncing contacts with Google, don't do the Personal Information Manager part. Stop when you get to the Restore step.

It would be wise, after the Transfer Purchases option to go through iTunes and make sure everything (media, app, book) got transferred. If not you may have a few more steps to insure all your stuff gets safely backed up on the computer.

NOTE: If you have picture albums in the Photos app, and they are not backed up somewhere other than the iPad, you need to save them first before doing anything else. The first sync will remove them. The PhotoSync app works well for this, and you can use it before starting any of the other procedures.

The quality of the pictures recovered this way will probably not match the originals. iTunes optimizes photos for viewing on the iPad when it syncs them.

Thanks Twerppoet,
I'll follow this procedure, much appreciated.

No problem. Let us know how it works out.