You need some oak edge banding.

How wide is that widest edge, 3/4" or 7/8"? Install (glue or iron on) whatever width you need to cover that widest edging. You might want to edge band the whole edging, not just the areas you sanded too much. You've sanded to raw wood, so the banding will/should adhere well. If you get glue on, you may need clamps, to clamp it until the glue dries. Elastic or rubber band type straps may be best for "clamping" along any curved profiles.

Home Depot has 3/4" wide edge banding.
http://www.homedepot.com/p/3-4-in-x-...2297/100541135

This place has 7/8" edge banding.
http://www.veneersupplies.com/catego...%22__x__10%27/

Sonny