Can someone help me?

I'm being talked about at work by almost everyone, and I want help in discussing with my manager about leaving early on my notice due to mental health, and how do I begin to apologize for upsetting people. I recently found out that I've upset a lot of people by not always saying hi/bye. I have trouble understanding social situations and don't always know when it's the right time to speak to a person. They are all closer to each other, so they often have their own conversations, and I don't want to butt in, so I'll usually occupy myself, waiting for my moment to speak, but then I get distracted and forget that I haven't spoken to that person. Also, I work retail, and I am usually a closer, so I'm preoccupied with doing a million things and not paying much attention to people around me. I often feel like I'm on autopilot, so I tend to come across as rude or standoffish. I'm worried about leaving earlier than my scheduled notice, just cause my sibling is still at the company, and I don't want them to face repercussions for my actions.

Please don't be harsh, because I acknowledge my behavior is not okay, and I feel sorry for making people this uncomfortable. I just want to apologize and move on. If anyone has any advice, it would be greatly appreciated.