Between 'seasons' is a good time to move to France. i.e. Spring and Autumn. The French take the summer very seriously, and largely very off! So everyone piles on holiday in early July to find the beach, or a campsite with a good lake / swimming pools etc. At the start of the summer the traffic can be awful, and at the end too. Don't move then!
Winter I simply wouldn't recommend due to weather considerations. Of course it does depend on where you're moving to in France, and the route you need to take to get there.
We moved to the French Alps, and practically speaking you simply can't move here in the winter due to the weather. It's fantastic to be here for the winter season, but I wouldn't suggest trying to make the move during the winter time. Roads can be closed, you might not be able to park near the house you're moving into due to snow etc etc.
We've driven from the UK to the French Alps in the summer time for holidays before moving here, and not hit any awful traffic at all along the way. Except for the one time early on when the sat nav sent us via Paris. Never again! But we have now got a great route and have always avoided travelling at peak times.
When I drove here to move house we came in March. That's still ski season but heading into Spring, which generally means the roads should be clear. I had chains for all four wheels in the car, and I knew how to use them! We arrived as a blizzard set in but just made it before we would have needed the chains.
Being prepared is key.
Work-wise of course holiday destinations have seasonal work, and perhaps very little work out of season. So if you think you want to move to a holiday destination permanently, check that there's summer and winter work for you. Where we are the town is pretty much empty May and June, and then again in September and October. No work really, so seasonal workers take their holidays at those times. There is lots of winter work, and quite a lot, though a bit less, in summer time. Many French people who work here every winter season (when most workers do 7-day weeks for the whole season end Nov to start of May) have regular summer season jobs elsewhere, often on the coast.
In May nearly everyone who lives in our town goes on holiday - taking their children out of school. The school deal with it as these people do need a holiday between working all winter and starting summer season work near the start of July. Plus we don't have a half term holiday between the end of the Easter hols and summer- though there is at least one week in May that is so full of public holidays it may as well be a week off!
Hope this helps.
Wishing you well,
Naomi