Hi, Max, and welcome to the forums!
I can relate to a lot of what you say. I also sat alone in high school during lunchtime near a corner. I made it through because I wouldn't look up lol. I would keep my mind occupied with other things. I also was reading, studying, writing, just doing something. Some people might approach you and extend an offer to have you sit with them. Take it unless you know for sure they are jerks. I sat at some jerk lunch tables that used me, made fun of me, and sitting alone was preferable to that. But if in doubt, take that offer to join others. There were some times I was approached and asked if I would join some people, but my knee-jerk reaction was "no" and I very much regret those decisions.
At least people notice that you look sad or upset; that shows they notice you and care at least on a basic level.
Have you spoken with your parents about these feelings?