And when you file for unemployment, your employer will not be the DOC, it is your contracting company that was providing services to the DOC facility. You were on their payroll, they were paying in employer taxes for you, and they'll be the separating employer.
And none of what you told us regarding how unfairly or badly you were being treated by the correctional officers on the worksite who are DOC employees is going to change that first thing you mentioned, that your company, the contractor, suspended you at the request of the DOC. The DOC alleges that you had an inappropriate relationship with an inmate.
If the DOC wants you out of their facility, for whatever reason, the contractor will do it. The DOC is the contractor's customer, after all. They'd rather please them than be fair to you, or even to give you the benefit of the doubt. Whether they made up a false allegation concerning you and an inmate, or whether they decided they just didn't like the color of the socks you wore to work that day, they can do it. They just tell the contractor, your employer that they do not want you there anymore. It's not an EEOC issue, it's not abuse, and your termination was not wrongful.
However, I agree with chyvan, the first minute you were put on suspension without pay, you should've filed for unemployment benefits. And week you are not working and not being paid, you may qualify for unemployment whether you've been officially fired or not.In fact you may not actually be terminated from the contracting firm. If not, and they do not have another work assignment to send you on, you likely are going to be approvable for unemployment benefits, regardless of the reason why you were suspended from your work assignment at the DOC. (Depending of course on your monetary eligibility based on how much work for covered wages you have done in about the last two years. )
If the contractor, your employer does decide you're terminated, in other words, they only have the one client, they're not going to send you on any other assignments, and they respond to unemployment stating that they have fired you from their business for a valid misconduct reason, related to the DOC complaint, then you'll have all that to go into with them in the appeals process. How you were treated by the contracting facility employees may or may not be a factor. The unemployment system will ask you what you were told by the contractor, your employer, and you will tell them. Then they will contact the contractor, and they'll make a decision whether or not the contractor had a valid misconduct reason to terminate you.
But regardless of whether or not you are approved for unemployment does not mean, in any way, that you are or are not the victim of a wrongful termination. And in the experience and estimation of the many people here who are familiar with hiring and firing issues, based on what you have told us here, you aren't. Many people have misconceptions about what the employer can and cannot do to them and how they are to be treated on the job, and they only find out differently after it actually happens to them.