Even without the use of new eggmoves as bellydrum azumarill, there was a way to break through these toxic stall gameplan.
Some common physical sweepers can destroy these Blissey/Chansey, Clefable and Miltank walls. Swords dance is an amazing wallbreaking move when using on pokemon which are immune to toxic or doesn't mind being paralyzed such as Excadrill and Scizor. If you have no way of beating stall, why not trying to use a stallbreaker in your team, using Taunt to shut down any wall relying on toxic/heal move? For instance, we have two amazing stallbreakers in the PRO metagame with bulky Gyarados who taunts any wall without an electric move to prevent them from healing/using toxic and sets up dragon dance on their faces since they can't hit it very hard. And of course, there is defensive Talonflame with Taunt and foremost Roost which keeps it healthy, boosting its average attack to sky-high level with swords dance and brave bird them to oblivion. It also helps against common sleep abusers since a lot of them are actually grass type.
And if you really struggle against status moves abusers, you might consider using a cleric (using aromatherapy/heal beal) in your team to deal with them. Hope it helps!