Welcome to KY. And congrats on the marriage.
Once you become a resident of KY, you can take a class with an instructor.
8 hours, open book 25 question test, and range test (11/20 rounds at 7yds).
Your certificate will be sent to your residence, and then you'll take it to the S/O in your county. They can take your picture, or require you to provide a picture. $60 is the cost for the license, and is good for 5 years. It also bypasses NICS as long as it has your current residence on it. You'll need to provide 2 payments to the Sheriff's Office---one for $40 to the state treasurer, and $20 the county to do the BG check. No FP cards needed.
If you move to another place in KY, a change of address can be completed at the S/O and won't take very long to update.
It will qualify you to carry a firearm, along other types of weapons. Death star, shiruken, billy club, slapjack, artificial knuckles, weapons of mass destruction, knives other than ordinary hunting knives (think OTF).
As far as your OH CCW license being valid once you change citizenship from OH to KY, I can't answer that, but the KY license should be in your possession within 90 days. You can put a gun in your center console or glove box in the interim, and KY is an open carry state.