You can begin using your husband's last name immediately if you want, including on your application forms. There is no rule or requirement that you change your other documents before filing. You do not need to await an EAD to change your other documents, either. You can do that by going to the various offices with your birth certificate, passport and marriage certificate. I express no opinion regarding your eligibility for a green card.
Your question needs to be answered from a few different perspectives.
First, you can use your married name on all of your applications/petitions for immigration benefits since you are legally entitled to use your married name.
Second, you will not be able to change your name with the Social Security Administration until you have an immigration document to support the name change i.e. Employment Authorization or Lawful Permanent Residence Card (aka Green Card).
Third, you can not use your married name for employment or taxes until your Social Security Number has been updated with the Social Security Administration, otherwise 1) your employer will start receiving "no match" letters for you and 2) your tax returns will not be accepted if your name and number do not mach.
Immigration Green cards Green card eligibility Employment-based green cards B-1 business visa Employment Authorization Document Paying taxes as an immigrant Immigrants and bank accounts Employment Social security Foreign and immigrant workers Spousal immigration Birth certificate Tax law