Yes it will. The point of Outrage is to KO something, right? You KO something with Outrage, a fairy switches in. You have to use the move so it'l say "It doesn't affect Azumarill!" and in that time the fairy has a free turn.Just because Dragon gained a new immune match-up does not make it a bad type. Sure if the opposing team has 3-4 fairy types then your dragon-type STAB isn't going to be very good. But the opposing team will likely not have 3-4 fairys. Dragon still has fewer NVE match-ups than most other types. And it will continue to be a good type for Dragons to use, just not as good anymore.
Just because flying-types are immune to ground-types doesn't make ground-type attacks bad. Ask Earthquake.
Also Outrage won't disappear, it'll likely stop its raging effect when met with an immune pokemon just as Thrash stops when used on a ghost-type. So theres no huge drawback to it other than the regular "it doesn't hurt the fairy type".