New Job Mess-Up

I started my new job at the marketing firm last week and already messed up. On Monday, I was supposed to send an email about the upcoming project deadline to everyone involved. But instead of sending it out early morning like instructed, I sent it around noon after lunch because I thought the deadline wasn't until later that day. Everyone was stressed, and I felt awful. My boss gave me a look but didn’t say anything directly. Today at work, my coworker Sarah pulled me aside to tell me she heard about the mix-up from another team member. Now everyone's avoiding eye contact with me because of this one mistake. I'm not sure if I should just apologize and try harder or if I need to step down from this project. Does anyone else ever have a similar blunder at work? What do you think I should do?