Transfer H1 and apply a new H1 require almost same documents. Same with H4.
However, you don't need wait your new H1 approved. You can work only you receive the receipt.
University need do same job for you to tranfer H1 or apply new H1 (for them, it is same). So they need do some previous work to let this position open. Some university are very slow, such as my university. They told me I need all documents at least 2 month ago. But some university can finish first part within 2 or 3 weeks. So ask your new university and be hurry.
However, you don't need wait your new H1 approved. You can work only you receive the receipt.
University need do same job for you to tranfer H1 or apply new H1 (for them, it is same). So they need do some previous work to let this position open. Some university are very slow, such as my university. They told me I need all documents at least 2 month ago. But some university can finish first part within 2 or 3 weeks. So ask your new university and be hurry.