What choice do you have? You can either go to Court and take your chances with the Judge or wait to get picked up in which case you will probably be locked up. The Courts are overlowing so it is doubtful that a Judge is going to be bent on sending you to jail if you have already paid your check and fees. If there are court costs ask for time to pay those. If you can consult with a good Ohio Criminal Attorney do that. The Court isn't going to be able to get their money if you are locked up. Good Luck!
I think you should hire a lawyer to have the warrant set aside. You might be able to go to the clerk, have your file pulled, then have it go to the duty judge who may require the signature of a prosecuting attorney or a judge, if it has been assigned. Also, there is always a chance you could be arrested when you go to the clerk's office to have your file pulled, and there you are with a live warrant for your arrest. I took this to the deepest hole, however, I strongly suggest you hire an attorney.
This answer is general and is not intended to create an attorney client relationhsip and is for informational purposes only.