I sell all my eggs to restaurants. My girls were laying almost 2 dozen a day, and that made for some nice sales. My market is Madison, Wisconsin, where there are some progressive chefs!
As for how long you can keep the eggs, I've done quite a bit of research. Duck eggs, if washed and coated with oil, will keep for 2-3 months nicely in a fridge,longer, if you're not looking for performance eggs!. One of my customers reported that he desperately needed some eggs and found some of our eggs in his fridge that were 5 MONTHS old and they were fine. This is an experienced chef. When I process the eggs, I scrub them, coat them with olive oil, and put them in my basement until delivery, which is usually a wait of 2 weeks.
Hopefully you can find somewhere to give away your eggs; maybe a food pantry as someone suggested. Asian restaurants/groceries might be able to sell them for you, or they might use them in their cooking. They will also probably ask you if you sell your birds, as I discovered! No!
Duck eggs are a relatively new phenomenon in the food market in the US and you might find it helpful to print out a nutrient analysis comparing chicken eggs to duck eggs when you offer them for sale. Also be sure to point out the sheer size of a duck egg compared to the chicken egg and the extreme value you get from the beautiful, yet humble, duck!
Good luck.