How to fix the top board on a low, long chest of drawers?

Try putting contact paper, then paint that. It's probably not the best thing, but that's what I'd do. As you said particle board is a mess. Also you could put a piece of plywood on top of the board, then nail it to the board, and hope it doesn't crumble. Then paint it, and varnish it or put some sealer on it. Good Luck!

Did you leave it for a week before attempting to sand?

Fixes that I've seen people do:

• put another piece of wood or a marble slab over the existing top; the top was built in such a way as to hide the existing top edge.
• sand down the top, even to the particle board. Seal the particle board with a mixture of poly and mineral spirits (shellac and varnish work, too), then refinish with paint or an opaque stain.
• put a mosaic on the top or tile the top.