We regularly drive down to the Pyrenees, usually either from Calais or Le Havre so not sure about from Amsterdam but it should be a fairly straightforward trip down into northern France.
We never go through/round Paris, always down to Rouen, Le Mans, Tours, Poitiers, Bordeaux and on down to near Bayonne when we cut across to join the Bayonne/Toulouse motorway. There is a new motorway coming down from Bordeaux to Pau as well but the toll is very expensive so we generally avoid it unless in a big rush.
If you have kids an ideal place to stop on the way would be Futureoscope just north of Poitiers. Very good day out and plenty of sites around.
Have a look at the Hull to Zeebrugge ferry as well as an option, another two hours easy drive from Newcastle and may work out cheaper. How much is the Newcastle Amsterdam ferry coming out at?
Handy little map here showing the free/toll motorways.
http://about-france.com/routes-from-calais.htm
Try to avoid travelling on the continental holiday start/finish days as well if possible.
http://about-france.com/week-planner-2012.htm
Have a good time
