I just did this for a friend recently so may be able to help.
Account you are transferring domain to has to be a fully paid for type, not one obtained via discount coupon or one where DH has paid someone a referral fee. If in doubt check with DH.
You will need to delete the domain from old account and then add the domain at new account and set up the website. You will lose all emails at old account and there is a brief period while you are setting up email accounts when mail will bounce. Good idea would be to start with a 'firstname.lastname@example.org' which will collect all emails while you are setting up the other accounts.
Check the following: