There are a few different things going on here.
First, it sounds like you're asking where to file to modify your agreement. This answer depends on where the original custody judgment was filed. Were you originally divorced in New York? If so, then New York probably still has jurisdiction over your case. If you and your son currently live in Ohio and you've lived there long enough under Ohio law, you can probably get the case transferred to the Ohio courts.
Do you need an attorney? Well, you don't HAVE to have one but it's always BETTER to have an attorney. If you can't afford it, there are alternatives, especially if you can get your ex to agree to the change in custody. Many states have a self-help system where you can file the paperwork to get the case transferred and do the modification on your own (without an attorney). Contact the court in your county to find out if they have the applicable paperwork. I recommend consulting with an attorney to make sure you're doing everything correctly if you go the self-help route. A lot of attorneys will either review your documents for a fee or they will go over the process with you at an initial consultation.
There are also document drafting companies that draft up the paperwork for you for a really cheap fee. I must warn you, though, that you get what you pay for. In the paperwork I've seen such companies draft, there are usually a lot of mistakes. They'll use a standard form for every single case, which means they put in irrelevant information for some people, not nearly enough information for others, and sometimes they just flood the documents with incorrect information. They also don't guarantee any of their work and don't have professional liability insurance so if things go wrong - it's your problem and you are the only one stuck with it.
If I were you, I would contact the Ohio State Bar referral service and find someone to help guide you in the right direction. If you don't find the right attorney the first time, keep trying. There are a lot of people out there willing to help - you just have to find them.
Sign up to receive a 5-part series of useful information and advice about child custody law.