They may be looking for a good reason to fire you, and have trumped this up just for that purpose. Forget it. you've made your explanations, and they have not been accepted. In your state, in the absence of a union contract, you have no appeal rights of any kind on the job and no labor laws have been violated even if they falsely accuse you of setting the building on fire or doing any kind of nefarious deeds in write ups or evaluations.
Do not let them make you angry enough to quit the job, because they may very well be trying for this, knowing you are much less likely to get unemployment benefits if you voluntarily quit the job, which would save them money.
So I'd take the hint, and cover my behind every way possible while doing my job to the best of my abilities. Meantime, be looking around for another job while you are still employed there. Best case, you leave and go to that new job and get away from this work situation. If they come up with something else that you have supposedly done, and fire you, you will file for unemployment benefits at once. It is their word against yours that what they wrote you up for actually happened and that they did have a valid misconduct reason to fire you. Best case scenario here for you is that you win approval of a claim and then go find another job, so why not be proactive? In this "at-will" state, they can fire you for any reason at any time. Your option when falsely accused and disciplined for something you did not do is to leave at will and find another job.

