I can say this from personal experience, if your boss is being a bully, it's better to quit. Mine waited until I screwed up and dropped me (so she could do it for cause) and she was like yours....talked over you and she treated most of her employees like idiots. Because I was new and learned diferently I was singled out, and she would talk over me and chewed me out on everything .I know you don't really want to quit but if your boss is lying about you and you think it is something HR can't fix ( it's hit or miss, I didn't go because I felt she's retaliate if I did) then you need to consider leaving this job sooner rather than later. I applaud you for standing up to tour boss, you are loads braver than I was! But there are better jobs out there. I looked on Indeed to find jobs because you hear from the people who work there, I don't know if that will help but you can see from what people say if the job will fit or not. You are not worthless and don't ever let anyone make you feel that way. Best of wishes.