EU resident work/live in different EU country?

I am a US citizen currently a resident of Spain, but I will not be able to apply for the same type of residency when this card expires. In a few months I will have been living in Spain for three years, so I was planning to apply for social arraigo in December (I understand and meet all the requirements). This will give me residency for one year. However, I have recently decided to look for work in France and I am almost certain I will not find work there before December. Can I live and work in France with my Spanish residency? Or will I still have to return to the US to get a work visa for France? I am trying to decide if it is worth the money/effort of getting social arraigo if I move to France in less than a year.

If anyone has any insight or can direct me to where I can find information about this, it will be much appreciated. Thanks