There are two parts of these work related visas. (1) Permit to live (2) Permit to work in USA.
H1B is a dual intent visa : It embeds both (1) and (2) above.
H4 is only (1) above.
Now the new rule adds (2) to the H4.
Having said that, you can apply for both, H4-H1 COS and H4EAD. Both run in parallel without one impacting the other. Once they get approved, you can use either of them. But, if your H4-H1 COS comes through FIRST, you will not be in H4 status. This impacts your H4EAD since the basic requirement for H4EAD is to have H4(and then to have an approved i140 for the spouse on H1).
To point the answer to your question, you can continue to work on your COS approved H1 (at that point, your pending H4EAD would be rejected)
I think ImmiGeek’s answer was on point and accurate. I would like to add to that. You are on H-1B presently and would like to apply for H-4 change of status and H-4 EAD. Nothing wrong with that, it is doable. However, applying for both (status change and EAD) at the same time worries me a little because simultaneously applying for EAD suggests you are assuming that your COS will be approved and DHS does not like such assumptions. Your COS approval notice must be appended to your EAD application for best outcome. Also, you must end work once the H-1 to H-4 change of status is approved and then wait for EAD issue before restarting employment.
Best wishes
Dr. Sandeep Shankar, Colorado Heights University, www.chu.edu; sshankarATchu.edu