The issue you’re describing, can’t use a Free Transfer because of Faction Conflict, is not uncommon. I’ve created a thread that could use your support as an affected party addressing it here: Classic Era Transfer Issue #1
Being an Alliance on Sulfuras, your only ‘free transfer’ option is to Whitemane, but as you stated, have Horde on the server.
The thread above has proposed 2 different but effective solutions to this problem. Once implemented they would be handled via the automated system.
They transferred to Whitemane but they disbanded shortly after. There are other active Alliance guilds on Whitemane (PvP West).
My Horde guild was on PvP East 1 (Benediction, Faerlina, Heartseeker, Incendius, Netherwind) but we’ve begun moving to Whitemane as well. We were the last active guild there. It’s now a dead zone. Whitemane has an active population and the horde are very receptive to new guilds/players.
For PvE Alliance and Horde, the free transfer destination is Mankrik. It also has active horde and alliance raiding guilds.