I will not be able to tackle of of the issues you have touched on and have no advice regarding any further steps you should take but let me explain some of the things you seem to be having trouble with.
1) There is nothing inherently wrong with using a public defender. Often, they are the most experienced criminal defense attorneys in a given jurisdiction. Attorneys in the know will tell you that the PD often has its fingers on the "pulse" of the courthouse. The PD is for the unable, not the unwilling. If you had been in a position to hire a private counsel, you certainly should have. You would have had the advantage of having your choice of counsel rather than an overworked attorney paid for by tax money. Running from a traffic warrant was a much graver error than failing to secure private counsel.
2) You were charged with a Class 2 felony because aggravated battery upon a police officer is not a Class 3 felony but rather a Class 2 felony per 720 ILCS 5/12-4(e)(2):
"Aggravated battery that does not cause great bodily harm or permanent disability or disfigurement is a Class 2 felony when the person knows the individual harmed to be a peace officer..."
3) I think something is being left out with regard to your sentence. You claim no prior felony convictions but give no indication of your prior criminal record or any record of performance of probation or other incidences of failure to obey court orders. Generally there is a presumption in favor of probation. The Court has to find that the presumption in favor of probation has been overcome to impose a DOC sentence for such an offense. If that was not done, your sentence might not have been appropriate. Presumably that was adressed at sentencing but you make no mention of sentencing alternatives or plea offers presented during the pendency of the case.
4) You have apparently now learned the importance of a person's credibility. What many people fail to understand is that even pictures, audio, video and other "concrete evidence" never stands alone. The value of such evidence ALWAYS hinges on the credibility of a witness that tells us what the "concrete evidence" means, when it was created, how it was developed or why it is relevant. Proof beyond a reasonable doubt has never meant "scientific proof." At the end of the day, the jury determines the facts by weighing the credibility of the witnesses and the value of any other evidence presented. In your case it appears that the jury believed the testimony of the officers and consequently found you guilty. If nobody could be convicted on the basis of testimony, nobody could ever be convicted.