My question involves labor and employment law for the state of: KY

I'm being relocated to another state. The company is giving me a one time, lump sum of $x,xxx that will be included in my next payroll check.

I don't want to move, but feel I really have no other opportunity within the company. Let's say I get settled into my new role and I'm not doing well. I ask to come back and they say it isn't an option, so I quit.

Seeing as I didn't sign anything and I'm an at-will employee, am I liable for any of the relocation expense?