As far as I can see, we're basically stuck with ICANN now. They got the IANA contract while managing to create a circular oversight process - where their board are only overseen by smaller committees made up of members of their board - and when they can't cover up the stink, they commission and independent report which they then submit to another sub-committee of their board and ignore it. See the .amazon or .aftica sagas for details and repeats of the process.
So they'll happily sit there and milk the bonuses and 5 star travel - and as long as they don't so totally fuck up that it's worth the effort of completely ripping up internet governance, we're stuck with them. Finding something better risks putting Russia and China in charge of global internet governance instead.