I hate going to the grocery store...and headphones do help alot. I totally think you should be proud of yourself for going!
By the way, I spent alot of my day during my last job crying at my desk (silently, the running-tears-down-face type) and I noticed that alot of people NEVER even noticed. Of course, I used my hair sometimes to block it and other times I did not care and anyone who walked by could see.
But I got along well with everyone and no one treated me as if there was something wrong with me, even the supervisors. I think a lot of people have so much going on in their own heads that they only notice so much around them, so don't feel too bad about crying and/or feeling overwhelmed. You are definitely not the only one. I agree with Nicks_Nose...was going to type more, but couldn't have said it better myself. Hugs to you.