Amazon filing non amazon sales for EU

I am a UK business that is VAT registered for both UK and France. I ship products from the UK only. I use Amazon VAT services for filing France VAT via my France account every month. I also sell on various marketplaces in France where I zeroed the VAT to buyers in France going by the export guidance on the government website. I am also not charging VAT using my French VAT number. I’m i correct?

And how am I supposed to account for VAT/Customs charges when I pay these on behalf of the customer? Thanks.