Southern France to Normandy ?

There is a direct night train from Carcassonne to Paris, but at this late date it might be fully booked. You also can travel by train during the day.

Take the train from Paris to Bayeux.

Your best way to see the Normandy beaches is to take an organized tour. For information on bus service, go here.

For Normandy tourist information, go here.

What do you wish to see in Paris? Do you need to be in Paris on 22 August?

I think the mistake that a lot of people make when wanting to visit Normandy beaches is thinking that it is a point. The beaches stretch for over 50 miles -- east to west. So when you talk about "walking the beach" you need to decide which beach? And then the areas behind the beaches stretch for another 15 to 25 miles so the whole area is massive and spread out. Second, by choice, it is an isolated area with no public transit. Part of getting the feel is not just the beaches but hedgerows and the areas behind it. You really need to take a tour to do what you want to do.

And I am not a big fan of text english.

Jane, you say you want to "walk the beach" in Normandy while feeling and being safe. Easy to do unless there is an invasion while you are there.

Are we to understand that you are currently in southern France?

I find it really hard to understand exactly what the OP is after because with the unusual sentence construction I get confused. I understand that you prefer to travel by water, and want the most inexpensive. Unfortunately that is a contradiction. The only ways to get from Carcasonne to Normandy by boat is either a cruise ship around Spain and Portugal - - very slow and very expensive, and I don't know of any service from that A to B; or through central France by barge. That would be really slow, and again expensive and indirect.

You could have a read of the book, "Skinny Dog To Carcasonne" for inspiration.