I wish I had good news for you but the situation is complicated as you may have already found out.
First of all, your priority is to ensure you are not out of status. It appears that your employer may not file your H1 application prior to the expiry of the 60 day deadline (starting from March 1 in your case). This means you will be out of status once your 60 day deadline is up.
Next, you are applying to a change of status to H4, prior to your 60 day deadline being completed. Since you will have a pending COS application to H4 status, you will not be out of status after the 60 day deadline is over. However, in the period between the expiry of your 60 day deadline and the day USCIS makes a decision on your H4 application, you are not considered to be in any status. You are considered to be in “a period of authorized stay” until USCIS makes a decision on your H4 application. This allows you to stay in the country, but you cannot work for any employer. In addition, H4 COS applications do not have a premium process available and typically take around 3-4 months to be approved (sometimes longer).
This “period of authorized stay” is what will complicate the new employers H1 application. By the time the new H1 application is filed, you will not have a status—you will be in “a period of authorized stay”. You won’t have a status to extend or change. So the H1 application will be approved with consular process—you won’t be able to begin working for your new employer.
If you have a valid H1 visa stamp in your passport, under normal circumstances you would be able to leave & re-enter the country with the existing H1 stamp. This would change your status to H1 automatically and you would be able to begin working for your employer. Your I-94 would be valid until the expiry of your existing H1 visa stamp. If you have a valid H1 stamp, you can explore this option.
If you don’t have a valid H1 stamp in your passport, you may have to wait until the consulates resume issuing visa stamps.
I ran into some of these issues; for me things got more complicated because H1 premium processing was suspended at that time. So I had to wait for months, had offers withdrawn, etc.
Good luck with everything and consult a lawyer. These are difficult times, but with some patience and perseverance, you should be fine.