Tort law

Did she lie? Did she provide falsified information to your employer? Remember that she did not fire you. Do you know for a fact that her intention was to get you fired? (Can you prove it?) Or did she simply report the matter to your employer? Your employer fired you. If they failed or decided not to investigate the information they received, that decision was on them. And yes, they can do that.
I am an animal rights advocate. I worked for the City Schools as a special needs paraprofessional. The county shelter director got very angry with me when I posted on social media that she was killing animals with no cause. Long story but the group I am over helped get the kill rate of the shelter from 80% kill to less than 2% kill in less than a year. She worked with us and we helped everyway we could. Suddenly she started killing animals even though she had room and no reason to kill. I exposed her on social media. She in turn sent my employer an email along with copies is several text conversations I had with her that happened to be during the school hours. Finally after looking at my attendance, out of the 8 she sent only 2 were during school hours. I have a copy of the email she sent using her county title. I’m positive it was sent from her county computer, sent during county time and using her county issued email.

Your post answers your question.

As previously suggested, don't go after powerful, wealthy people.

They taught this dummy a lesson.

I know my place and stay away from my betters.

Whatever our betters do or say, its none of our affair.
Okay, let's cut to the chase.

Based on the information you have provided, the answers to your questions I am trying to figure out if there is any legal avenue I can pursue against the county OR sue this person directly for what she has done to me/my family are No and No.
I suspect their decision was based on more than this single incident. They may have had other complaints from within the school, or already made their own observations of you engaged with your phone or other devices. Many people are easily distracted with their access to social media these days.
Have to agree that you (1) poked the wrong person and (2) used the wrong way to go about complaining about the animal issue. You took on someone more powerful and lost in a consequence you weren't expecting. In the end you gave her the ammunition to use against you (texting while at work for non-work-related issues that might have been seen as harassment -- I didn't read the actual text messages you posted) If nothing else your mind, even if you texted while on break/bathroom, your mind was still somewhere other than on supervising the kids.
Trust me when I say I know of what I speak; if you are a special ed para and you are on your phone and texting about non-work related matters when you are supposed to be supervising students, it is a huge deal. That others may sneak in a text or two and not get caught does not make it ok or excuse it. Your phone shouldn't have even been out and a distraction when your attention should have been focused on the student or students you are assigned. I can promise you that you are not in the only school system that would have acted the same. If this is a public school, then you would have certain due process rights, though that would only apply to your termination with your employer.

Quite frankly it it really wouldn't matter one bit if it were the Queen of England who reported you or a clerk at the local 7-11. Put it this way, if you were being criticized at work for how you did your job by a parent, would you not expect to be able to use your work email to respond? Why would you think she would have to wait until she got home at night and was using her home email to report that she was getting texts in the middle of the school day from someone who should be working with vulnerable children?