Any seed that has a sizable layer of gelatinous arial (like Ginkgo), I never allow to dry a lot or completely. (In some cases, e.g. crabapple, I have found it near impossible to get the real seeds out when they are fully dried up.) As you found out, it is a royal pain to clean them when dry. I clean these right after they drop. And yes - you better have a good gag reflex.
I clean them by rubbing them over a large sieve I made ~ 2 ' x 4 ' out of 3/8 " hardware cloth mounted on 2 x 2 structure. Works fast and great. I also use this on may other types of seeds as well - most recently paw-paw. I have 3 of these sieves with different sized hardware cloth. I also use these to sift soil - works great to remove rocks.
I always wear very heavy kitchen gloves like Rubbermaid and keep them on the whole time.
