Once the domain name was successfully moved to the new registrar, however, it said I couldn't register these nameservers again because they already existed. For now the nameservers still work, but I'm afraid that one day down the line the original registrar will delete them without warning.
Do I need to tell the original registrar to delete these nameservers and then create them at the new one?
If in doubt, contact your new Registrar and voice your concerns; I suspect you'll get a reassuring answer back, which should put your mind at ease.
R.