Have you been ever looked for a trustworthy person to transfer gold in world of warcraft? Don't worry about it ! There is a brand-new free transfer service for all of you----game player who are interested in making quick,easy and safe trades.
Well one way of doing it is offering to give someone all your gold on your server if they give you the gold on the new server. I know of some of game players who have done that and they did not get ripped off. Just advertise in Trade that you are looking forward to trading gold between the servers and make sure they are in a respectable guide. There is a percentage gold change depending on the servers you choose, so that you can grow your supply to offer everyone.
There is no way to directly transfer your gold from your Alliance characters to your Horde ones. You are correct that you cannot bid/buyout on your own auctions in the neutral AH. The usual workaround is to get the assistance of a trusted third party who has a character with neutral AH access.
 You will set the price for more than money than you are wanting to transfer The Neutral AH Takes a hefty chunk of the sale price in admin-costs.