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There are two types of custody. There is LEGAL and there is PHYSICAL
And so she could have sole PHYSICAL custody and you two could share the LEGAL custody. Having the legal custody means that neither of you can make life changing decisions that are not an emergency situation without the consent and agreement of the other. So for instance, the 12 year old comes home to her and says "I want to get a tat" AND she says OK. That cannot happen unless she first discusses it with you and you agree.
Now, if she has sole physical custody with the shared Legal custody, then you will be responsible for more of the child support. If you both share Legal and Physical then she would be responsible for some share of the child support.
And she is correct, if can be stipulated that she cannot move away with the children farther than a specified radius from your home which is generally 50 miles.
So if you are going to agree to the Sole Physical to her with the share legal, then I suggest you would want to have overnights in your visitation plan. Such as Friday from 6 pm til Sunday 6 pm - every weekend, every other weekend.
If you want joint custody, then tell her - you want to share the physical and the legal and that you will have the children from Sunday at 6pm until you drop them to school on Wednesday and then she has them until Sunday
Also work in the summer vacations, holidays, Mother's day she gets them, Father's day you get them with all birthdays alternating even and odd years.
So as to your rights - if she has sole physical - the only right is not seeing the kids as much as you'd like.
But as long as you share the LEGAL custody, all your parental rights remain
I hope this has been helpful. You can respond here if you need more clarification.