First : You are applying for H1B visa(petition) at this moment, why you are getting worried about VISA stamping(which comes way later in the year)?
First take copies of your current passport and get your H1 papers ready for filing.
Then immediately(dont wait till your H1 filing), file your Passport renewal. If you get your renewed passport before March ending, supply the new passport copies to your H1 filing company. If not, dont worry.
Last but best is … dont think too much about stamping of VISA at this moment.