If you open a Natwest e-isa and fill out the online transfer form the transfer will all be taken care of for you. The funds will be sent from your current ISA provider to Natwest.

Having paid into the old isa in this tax year will make no difference and any remaining subscription should remain unaffected.

It worries me that you have said "can I now transfer the balance to a Nat West e- ISA and open a new Barclays Golden cash ISA?" You cannot operate 2 Isa's in any one tax year. Had you not have paid into your old ISA this tax year then yes, you would have been able to

However, as the subscription has already been started and would be included in your transfer balance it would not be permitted to have another ISA account.