What do states do with wrongly convicted?

Your term "Wrongly convicted" covers a whole lot of different areas. Each state has its own procedures. At the trial a person is found guilty or not guilty. (In Europe there is also a verdict of Innocent.) There is a possible appeal. After appeal review, and sentencing, the conviction is final. A person is then convicted.
After that post conviction relief is possible. If new evidence is discovered that could have been presented in a trial, the prisoner submits a petition to the trial court. (It has a technical name. There are many possible petitions. See a lawyer for your state.)
If the judge goofed up, there are motions to the appeal court. (See a lawyer for the technical rules for your state.)
If all else fails, your state has a provision for a pardon.
The various state legislatures have generally made provision for payment for the time spent in prison for persons wrongfully convicted. (As a result, Judges will sometimes have a person plead guilty to a small crime in exchange for time served!)