The more I read about public defenders on this and other sites leads me to conclude that they are overworked and underpaid and, yes, I do think they want to get through their cases as fast as possible without having to go to trial.
That's entirely up to you but this layman's two cents worth is that the charge is unfounded. I would urge you to consult a private attorney even if you can't afford one. Initial consultations are typically free or low cost for a half hour or so just to get a second opinion.
Yeah, they do make mistakes.
Give yourself a little more credit. You express yourself rather well here.
No need to be afraid of going outside, just don't run so fast anymore.