Wood is very absorbent, and will absorb resin from the laminate before it cures. What you should have done is give the wood a light coat of epoxy resin (some people thin the resin with isopropyl alchohol, but I don't bother), and allow that to dry for 12 hours or so. This seals the wood and you can then apply your resin and glass over that.
Sadly I do think you will have to sand it back and start over if this is going to be cosmetic.
For the areas with white lines, you should be able Dremel in and backfill with resin.
grease or other contaminates tend to cause fish-eying rather than white patches. Also if you laminate on a day with a lot of humidity that can leave a milky finish in the resin. I tend to put epoxy work in a hot cupboard, or a hot box. It cures really nicely that way.