Not enough information yet.

1)  How old are you?

2) are you in college full-time or at least half-time?  (undergraduate? Grad school?  other?)

3)  "can" your ( OH? )parents claim you as a dependent on their tax returns ?

I am 22, a full time college student senior status, I have solely supported myself in 2019 so neither parent will be claiming me. 

You'll have to be ready to prove you supported yourself for  2019....the IRS may have questions....and make sure you talk to parents about it first to make sure they are not claiming you, because self-support for someone your age and going to college is unusual.  (I know some of the other SU's have a link to the self support test page.   Whether they "are" claiming vs. "can" claim you for 2019 are two different things.  You need to show you supported yourself for more than half your living expenses for 2019.

_______________________

IF you are continuing down this path...and Beyond that, if you have moved to KY, and are not a dependent of either of your parents, you would indicate you moved to KY last May from OH on the "my Info" page  (your mailing address is immaterial, but you should have one now in KY where you expect to get your mail for the next months, and file change of address forms if you move after filing your taxes.  ALL income you had before the Move is OH income, all income after the move is KY income.  You will have to delineate those sub-portions in each state's part-year tax returns....but only AFTER you have filled in the Federal tax return completely.

Delaying to get the KY Driver;'s Lisc is more a procrastination issue in your case...you need to get that done, register and re-plate your car in KY, shift-to and pay KY car insurance, and register to vote there. 

If it is a state college you are going to, whether you can get in-state tuition depends on that particular state's rules...some require a couple years of in-state tax returns first..but you'd have to check at your college Registrar's office to see if you "might" qualify

For state income tax purposes, you are an Ohio resident until you establish residency in Kentucky.  Here is Kentucky's definition of "resident":  

"Resident" means an individual domiciled within this state or an individual who is
not domiciled in this state, but maintains a place of abode in this state and spends in
the aggregate more than one hundred eighty-three (183) days of the taxable year in
this state.  https://apps.legislature.ky.gov/law/statutes/statute.aspx?id=47361

Your "domicile" is your main, permanent home.  A taxpayer can have only one domicile at a time.

what kind of proof would I need to provide for self support? 

@aleach15 --  If your parents do not intend to claim you as a dependent, there is no need for you to prove self-support.

I would think the self-support part would be required, if the individual is going to claim a change of state residency....but that's more of an opinion, than any legal requirement that might exist (and we can't give legal advice).    Not sure either of the states would pick up on it right away though...except that KY sees you are living and working in KY, without being a dependent of your OH parents....that implies you are independent and they are going to be pushing to get their taxes from you for  after you moved out of OH.....and be on your case for not having a KY drivers Lisc, and vehicle registration/tags/insurance minimums.