Do I have to give up my rights if I move out of state? I live in Ohio and I am somewhat forced to move out of state and my ex is telling me that if I move I have to give up my rights as a father.

No you don't give up your rights.
You will have to go get your visitations modified but there is no reason to give up rights. How far away will you be? How old are your kids?
Very likely you may be able to have visits for summer vacation and Christmas

Maddie is 3. I am going from Ohio to NY about 3 hours away. Depending on my new job hours, I would be able to visit about every other weekend.

Then you should be good. Just go get visitation orders modified if you have to.
The Judge can even order that they drive half way there to meet you.