I have no intention to offened anyone by taking this question.
You should do two things at the same time: 1) revoke/cancel your H4 application 2) file H-1b transfer. You can start working after filing the H-1 transfer.
But be very careful for this: DO NOT ASK FOR H-1 EXTENSION WHEN YOUR NEW ER FILES THE H-1 PETITION. You cannot ask for extention because your old H-1 status was broken.
To avoid the problem of both H4 and H-1 pending in USCIS, I suggest you wait until you get the approval of H4 before you file H-1 petition. You can start working immediately after you file the H1. So you will not lose much time.