I can relate to what you are saying. I have been lifting weights for about 6 years now and it hasn't improved my mental health at all. I have taken up cycling, running, and rowing all have resulted in no improvement.
But physical fitness has more benefits than just mental health and you should never give up on trying to improve yourself.
I have mild Autism Spectrum Disorder (ASD), as you mentioned in your post that you relate to Sheldon Cooper, ASD might be something to look in to.
As I said before never give on self-improvement - you will eventually find something that will help.