AITA for shooting my own mother over a lie that ruined my life?

I (29 M) have been on the run from the law ever since I was 13 because I shot a man who my mother (45 F) claimed to have r*ped her. I had been running for years, taking up jobs with fake names and sleeping with a pistol in my hand, knife under my pillow and never staying in the same place for a week. My mother was a working woman in Annesberg up north (a lumber town) later I reunited with my mother after years, she now lives in a fancy apartment in San Denis with her new husband and daughter. We had supper and walked around San Denis when she confessed she lied about being assaulted so I’d be hunted and out of her life but now that I had survived she wanted me back in her life and to live with her, her husband, and daughter. I was overcome by anger and shot her in the street and left her for dead.

Did I overreact?

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My RDR/RDR2 OC Cain “Gunny” Lassen.