Congratulations on your new love! Moving out of state depends on several factors regarding custody. If the father of the children wishes to continue seeing the children, then he can fight you in court to keep you from taking the kids out of state. If you choose to take your kids across state lines without legal permission, you can be charged with kidnapping even though you have custodial rights.
You have several options though. If your ex-husband wishes to continue seeing your kids, then it is likely you will have to carry the burden of travel expenses if you choose to move out of state. You could agree to pay these expenses and move out of state. If he does not want to keep seeing your kids, and depending on how serious you are with this new man, he could legally adopt them. Be aware that this will end any child support agreement you have with your ex-husband. A third option is to minimize travel time for your kids if you move to another state by having their father take them for all of the summer, and you keep them during the school year. This is difficult if you like family vacations, but less of a financial burden.
Your profile says you aren't getting any child support. If your ex-husband owes child support and isn't paying, then you could offer to forgive past debt if he gives up legal rights to the children. I personally would not recommend this route, as it also sounds like he treats your children well.
I think Stacey's suggestion is a good one. Could this new man consider moving to Ohio for you?