Note - we are all providing non-expert legal advice.
You should be good, Ohio doesn't fall under category 4. OH doesn't recognize most weapons as FL does for conceal carry, only firearms.
Many places offer getting the FL license in lieu of OHs. Many others recommend still getting OHs as well.
Whereabouts are you living in OH? There are places everywhere that have CCW courses. They would clarify a little better for you.
I am a resident of Ohio and have a FL permit. At the time they(FL) had quite a few more reciprocity states. That has changed a lot now though.