1) you need to apply for COS from H1 to H4, if you are currently on H1. You cannot use the EAD until the COS is approved. Onluy when COS is approved you would officially be on H4 status and then you can use the H4 EAD.
2) I would suggest that if you have a valid H4 visa stamp on your passport, you can resign from H1 job, exit the country and re-enter the country on H4, which converts your H1 to H4 instantly as you enter, then you can start working on H4 EAD.
COS generally takes weeks or months to get approved.