Having now received my Withdrawal Agreement (WA) residency card, can I travel throughout the EU using this card or do I still need to carry my British passport for this?
As a foreign resident of France with a residency card, you can travel freely in other Schengen area countries for up to 90 days in any 180-day period, but paperwork is still needed.
You should always have your residency card with you in France in case of a police check, and this card can also be used as an identity document for internal flights in France.
You should take it with you on trips outside France within the Schengen area, though there will be no automatic border checks if you are not travelling by plane.
In this case, however, you do also need to have your passport with you, as the residency card is not sufficient on its own.
Both documents will also be checked if you enter or leave the Schengen area – for example, if going from France to the UK.