It is against the law for an employer to retaliate against an employee who either asserts his/her lawful rights in the workplace and/or files a complaint with the appropriate federal or state agency. It appears that this might be the case with you.
These cases are complex and thus I would strongly state that you should see an attorney.
Offhand, I would be surprised to see an attorney take this on a contingency fee. TO go forward, you will need documentation, particularly *documenting* that you were "falsely accused."