As a French national, you would be best advised to try Guadeloupe or Martinique, or St. Martin.
Don't bother writing. Take a copy of your CV and arrive on their doorstep. Arrange to be out there for two or three weeks, and make sure you stay somewhere they can contact you after a couple of days or so - taking your mobile phone and buying a local pay-as-you-go SIM card is also a good idea.