You don't need to "transfer" H4 -- your H4 doesn't care about which employer your spouse is working at. However, you need to *extend* your status as your H4 status will expire in a few months.
As far as I know this is not true. Automatic extension of work authorization while the EAD renewal is pending only applies if you have a valid H4 I-94. It may be that your H4 extension of stay will be approved before your I-94 expires in December, in which case you will have automatic extension of work authorization if your EAD isn't approved yet.
If not, one solution is, if you have a valid H4 visa (or if you are a Canadian citizen), you can exit and re-enter the US, presenting your marriage certificate and your spouse's new H1b petition approval. You should be admitted to H4 status with the same expiration as your spouse's H1b. Since you will have a new H4 I-94, you will have automatic extension of work authorization while the EAD is pending.