What would I be giving up if I agreed to give my ex-wife sole parental responsibility of the children?

If a parent gives up shared parental responsibility of the children, that means that the other parent has sole parental responsibility. The parent with sole parental responsibility can make all decisions regarding the child to the exclusion of the other parent. Decisions regarding health, education, welfare, religion, even where the child will live, even in a foreign country or a different state. You give up the right to share in the decisions regarding your children's lives when you give up shared parental responsibility. The other parent has sole parental responsibility.