According to Dave Meltzer, Dustin Rhode gave notice on January 19, which the company granted, though they did install a 90-day non-compete, which ended last week. The announcement that Rhodes would be facing brother Cody at AEW Double or Nothing came just after that period ended.
The Observer notes that Vince McMahon was against releasing Rhodes, only for Triple H to talk him into it. Given that it's Rhodes, it was almost a mortal lock that once he was free from WWE, he was probably going to be joining his brother's endeavor, in some capacity.
Meltzer also adds that, similar to the current Luke Harper contract situation, WWE could have added on more months to Rhodes' contract, due to the time he missed recovering from recent knee surgery. However, thanks apparently due to Triple H's intervention, that didn't happen.