I picked off a piece of broken fingernail. I was not sure what to do so even though it happened on a walk to the grocery store, I then bought a bag of carrots (they were wet in the bag if that makes it even worse). I washed my hands before eating them with my hands, but was it okay to eat them even without washing hands between the nail coming off and touching the outer bag when buying them? I worry I should have gone home to wash and then gone back out to the store.

shalom uvracha, In the Shulchan Aruch (4:20) it Is mentioned that one who cuts his fingernails must wash his hands afterwards. It continues, that the person should try to do it quickly because there is a "bad spirit" that comes from the cutting of a fingernail. However, there are a few reasons that can be mentioned as to why you did not do anything wrong, according to Halachah: 1) You did not touch the food, but rather just the bag. There is no problem touching the bag. 2) Even if you touched the food, the food is still not forbidden. 3) According to the Rambam and others there is no 'bad spirit' in our days. 4) The Chazon Ish holds that if one picked off or bit off a fingernail merely out of a habit, or any reason other than actual hygiene, he does not have to wash his hands. 5) According to the Ben Ish Chai (Rav Yosef Chaim of Bagdad) the 'bad spirit' is mainly referring to fingernails that are longer than the skin. Therefore you did nothing wrong and you didn’t have to go home to wash your hands. (even if you did need to touch the food). See Piskei Teshuvot (chapter 4: page 54) All the best