We have just done this. We had put it off as something necessary but dreading the process. After talking to someone who had recently done it decided to bite the bullet and get on with it.

Actually very straightforward. Go on to

administration-etrangers-en-france.interieur.gouv.fr/particuliers/#/

Select Je déclare un changement de situation.

You can then enter the details of your current TDS, create a login and upload proof of your new address.

You'll receive a confirmation email of your request.

Then about two weeks later you'll receive a message in your personal login (notified by email) to say that your dossier has been accepted and you'll be notified when ready for collection.

Finally after about another two weeks you'll receive notification (by text this time) that your card is ready for collection.

I applied online on June 28th, was notified of acceptance on July 10th, got my text on Monday and picked my new TDS yesterday from the SP in Beziers. Nothing to pay (at least in the Hérault).

All the dates on the TDS are the same as the original. The only change is the address.