My recommendation is when buying a new Mac to ALWAYS get one with the same or more storage as your previous Mac. Based on your description of the problem it seems that you did not do this. That means your ONLY option is to return the new Mac and buy one with the same or more storage as the old machine. Then your migration will go extremely smoothly. If you don't do this you will not be able to use Migration Assistant to migrate as it is an all or nothing process!!!
If you think your new machine has the same amount of storage (or more) as the old Mac, then please be specific and tell us how much storage the old and new machine actually have!!!
When you are ready to migrate again, please follow the steps in Setup Assistant and Migration Assistant.