First of all, read Article 20 of the French/US tax treaty regarding teachers and researchers. It states that your income from personal services from teaching or research is taxable in France for a period not exceeding 2 years from the date of your arrival in the US.. An individual shall be entitled to the benefits of this paragraph only once. If you have already claimed, you cannot claim it again.
As far as your H1B status, read this IRS source regarding H1B visa status and click on the section where it addresses non-resident aliens. If claiming treaty benefits, H-1B aliens must report both the income and the treaty benefit on Form 1040-NR, with Form 8833 (if required). Please refer to Claiming Tax Treaty Benefits.
If you are filing as a US resident, please read the the section regarding Interaction with Income Tax Treaties for further information a clicking on the source link in the previous paragraph,
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