My current lease has one price for pet fees. This is different (lower) than what is advertised on my rental unit's website. My landlord wants me to pay the higher price and won't agree to an addendum of the original price. Can they do that? They have tried to argue that I agreed to this price for my renewal lease (that does not start until next year). I am in Ohio.
Unless there is a unilateral change clause in your CURRENTLY effective lease, then no they cannot change the terms until the NEW lease becomes effective.
Your question, however, is not entirely clear. You seem to be asking:
"My current least charges me $X/month for a pet and the new lease, starting on 1/1/2023, charges my $Y/month for a pet."
In that case it's perfectly OK since it's a new lease that replaces the old one and it's entirely up to you to either agree to it or find another place to live.
Ohio also has prohibited rent control and rent stabilization state-wide (Ohio Revised Code, sec. 5321.20).
It depends if you are in a rent control area.
Outside of a rent control area, the landlord is free to set prices as desired, and the only restriction is a lease of some kind. So the terms of the lease control. If a supplemental fee is not stated in the lease, I see no reason it can't be adjusted at the landlord's will.
If there is a lease and it expires, the tenancy lapses to month-to-month with no price controls. This is why people in non-rent-control areas like to sign new leases (even though it's not required); it locks in the rent for the term. The landlord can't do a functional eviction by tripling rent.
In a rent control area, the rent control generally applies to all parts of the tenancy of the unit proper. So I don't see a way they could raise it, unless local law carves out "pet surcharges" as immune to rent control.