Nellie, I am truly sorry. I know this is something you were anticipating. You were as you say falsely accused, and this was the reason they gave for terminating you we assume? Honestly, from what you have said here, I think I would go immediately to a good labor attorney. Forget the union attorney, it doesn't appear that you and he are hitting it off or that you feel like he was helpful to you in any way. Do not think you can force him to become more helpful, or that you can self represent and put your own defense on while he's standing around supposed to be representing you.
From the sound of things, you may have some kind of case against the school system. You need to talk to someone about it, very quickly. You also need to file a claim at once for unemployment benefits. It may be a very small claim, due to your having a part time job only, but you need to get that established very quickly. What exactly does your termination letter say? Are they using the incident of your not being on the job as their only reason for termination?
DO NOT go into all this information you've been providing us about how you can show your son wasn't covering for you because he was somewhere else yada yada, just be very brief in your initial claim filing. Say you were terminated because you were accused of not being on the job on such and such date. Say this was NOT true and that you had denied it to them. Oh well, I'm completely shooting in the dark here, because I have not read your termination letter and do not know what happened. But in any case, I do believe you do need an attorney at this point, not the union attorney whom you have mentioned before as not being supportive.