Good news, I've got rid of the chips. I used my router, and there aren't any chips now. I took off tiny amounts and moved it very slowly. There are a lot of rough patches but they are with the grain and I don't think they could be avoided.
There are a few burn marks too, but I can't feel them so hopefully these will sand off easily. It's very hot today, so that probably didn't help.
How smooth should it be after sanding? Should it be completely smooth, should I still be able to feel the grain or does it depend on what finish I want? I'm hoping to have a lacquer finish.