We were at St Cyprien which is the next town along the coast 2 years ago when our girls were slightly younger than yours.
Collioure which is 20 minutes down the coast is worth a trip (we went a couple of times for lunch and then the girls had a splash around on the beach) although parking can be difficult. We were there at the end of June so it would have been busier.
The old part of Carcassonne itself is worth visiting. There's a big car park just outside the walls and it was ideal. Our older daughter loved going into the castle. Villefranche is a mini Carcassonne.

At night time Argeles Sur Mer comes alive as opposed to Argeles Ville which is dead

there is one pedestrian street full of bars and restaurants with live music and also street market if you walk slowly take your time drinking then you can watch the street change as market stalls close and new ones open up

one day a week there is an artists market in the place by the roundaboutArgeles photo's here - not many but the traffic island is a good meeting place as it's near the HUGE car park which was free, last time we visited in '05