Can I start the process while waiting for the new passport?

Although not clear from your question, I presume you are a U.S. citizen who has become married, and who now wants to sponsor your husband in a marriage-based adjustment of status application process. If that is accurate, then the fact that you are awaiting receipt of your new U.S. passport showing your married name will not stand in the way of eligibility. Of course, there are many other details that impact eligibility. There really is no substitute for you and your husband consulting with an immigration attorney who, after learning all of the relevant information about you and your husband, could advise about immigration eligibilities, options and strategies, and who then could offer legal representation in the often quite complex application process.