You take what you can get when you ask a question over the net, so here's my guess.
As soon as you are no longer a resident of Oklahoma, you're no longer allowed to CCW with an Oklahoma permit. And if you're caught CCWing in Ohio, and you present an OK license, the LEO's liable to say, "Well, son, this license is for Oklahoma. You don't live in Oklahoma anymore. You have the right to remain silent...."
You get the picture. Ohio LEOs are not the most tolerant I've ever seen.
Plus, depending upon what part of the state you live in, be advised that you might be required by local law to register each and every one of your handguns with the local police. I was considering moving to Toledo and discovered that fact. IIRC, it was going to be like when I moved to Michigan, where I had to physically carry all of my handguns to the courthouse and a civil servant physically registered each of my handguns.
Weird.