how Do I prove unauthorized use of my signature?
Proving a negative (here, the fact you did not sign the disciplinary documents) is usually very difficult, if possible at all.
Instead, most likely your proof would consist of circumstantial evidence that discredits the employer's allegations. To that effect, you would need to subpoena the employer for records and take depositions of individuals who are involved in the matter.
Subpoenas and depositions lead to a large portion of evidence that ensues during discovery proceedings in a lawsuit. Even if legislation does not require the employer to produce a copy of your file(s) upon your request, the employer could be ordered by the court to produce a copy of these and other records (provided that the judge does not arbitrarily quash your subpoena, and that it does not fail to enforce its orders).
would the burden of proof fall on the company?
No. The employer would produce (file in court) a copy of the documents he alleges you signed, and it would be your burden to prove that your signature was falsified.
what reparations can I pursue for having lost my employment as a result of unauthorized use of my electronic signature?
Cause and method are different things. The unauthorized use of your signature is not what caused your termination. Based on your description, it was just the employer's method to force your termination under an appearance of legitimacy, possibly with intent to cause you unwarranted harm. That being said, you might be awarded lost wages.
If the pretext for your termination is traceable to a false statement of fact, you might also have a claim of defamation, for which some jurisdictions purportedly award exemplary and punitive damages if certain factual and procedural conditions are met. See Sky v. Van der Westhuizen, 136 N.E.3d 820, 834 (2019):
The law of Ohio is clear on when punitive damages may be awarded:
[P]unitive or exemplary damages are not recoverable from a defendant
in question in a tort action unless both of the following apply: (1)
The actions or omissions of that defendant demonstrate malice * * *,
or that defendant as principal or master authorized, participated in,
or ratified actions or omissions of an agent or servant that so
demonstrate; [and] (2) The plaintiff in question has adduced proof of
actual damages that resulted from actions or omissions as described in
division (B)(1) of this section. (Emphasis added.) R.C. § 2315.21(B).
Interestingly, Ohio case law employs a different meaning of the term actual malice (compared with most other jurisdictions). See Sky at 834:
Actual malice, necessary for an award of punitive damages, is (1) that state of mind under which a person's conduct is characterized by hatred, ill will or a spirit of revenge, or (2) a conscious disregard for the rights and safety of other persons that has a great probability of causing substantial harm.
Your claim of defamation would be against the person(s) who uttered the falsehoods, not the company.
The Ohio penal code might also sanction the falsification, although it is hard to assess the chances for the attorney general to prosecute the person(s) who falsified the records.