Can I continue working on F1 EAD between H4 status approval and H4 EAD issue. Will I have to take a break when H4 status gets approved until H4 EAD is issued or can I continue working on F1 EAD until H4 EAD gets approved.
No, you should NOT. Yes, you need to take a break after H4 approval. As soon as your COS is approved, your H4 is valid and you cannot work on F1 visa EAD, while you are on H4 Visa.