Hey Katie, welcome to Res! =)
If you are finding eggs you don't want, not picking them up is a good option. There is a limit to how many you can pick up - 100 in a 24 hour period (roughly).
Another option is to sell the ones you don't want, or put them up in an item trade and see what kind of offers you get.
A third option is to just keep the extra eggs, and use them to turn into the Egg Quest, under Games. You get TU (the game currency) in exchange for turning in the eggs the quest asks for.