Thanks for the encouragement everyone! I spent most of the day trying to not sand the veneer. It is definitely very thin. I think I can see an area where the glue that hods it on is thicker than the veneer!
Marcel, thank you for the hint on the neck I did wonder how to cope with maybe true-oil on the neck and a stain or paint on the body and glue in- between.
Like a number of people have indicated the neck centre line does not quite line up with the body centreline. But I've never thought to look at that on any guitar I have ever bought or played, so I don't think it will worry me too much. I think I can get it to look straight and end up with the bridge on the centre line of the body.