I want to know if is it necessary to change my name on the passport after marriage. I am getting married in February and my spouse is a French citizen. He will do the paperwork for me after marriage to get me along with him. So, do I need to have a passport with my husband's name for that purpose, as my current passport is not expired yet?
No. Your marriage certificate will list your maiden name, and that's what is used for the paperwork. In France, you can be called by your husband's name (or not), but most official paper work (French ID card, driving license) will document you as Mrs XXXX (your family name), épouse (spouse of) YYYYY (your husband's name). Others, like the Carte Vitale (Social Security card) will list you as Mrs YYYY.
Depending on your country, when you renew your passport, your country might let you (or not) add your husband's name on the passport. Chances are though that they'll still want to list your birth family name.
My advice would be, in the specific case of France, not to change anything on your own country's ID documents -- as your marriage certificate will list your birth name -- and get your French ID papers (resident card) with both names listed.
(Source: been there done that.)