Your will is administered in the state in which you die. If you die in Florida it will be administered there, even if you are an Ohio resident. If you have property in both states, you will also need a representative in both states.
The will is administered in probate court. A probate court will usually let the executor handle most of the estate settlement activities, unless there is a problem.
Probating a will, especially for a complex estate, can take some time. There will be court fees and there is usually a clause in the will directing those fees to be paid out of the estate. If there is a will contest, the proceedings can get lengthy, messy, and expensive.
The probate of the estate is completed when there are no further questions or transactions possible under the will, and the executor files a final accounting.