As you have worked only in France, you are receiving a pension only from France. Your family benefits will therefore also be paid by France, under its own rules. The fact that you've now moved to Spain with your family does not affect your entitlement to benefits from France.
NO — You can't receive family benefits twice over the same period and for the same family member. There are "priority rules" in cases of overlapping entitlements:
However, if family benefits from the other country in which you are entitled are higher, that country will have to pay a supplement so that you receive the highest possible amount.