Here is the full list of all 14 Mollycoddle Mayhem achievements.
Dispose dirty diapers in bodies of water
Finish the Final Stage
Achieve 3 Stars in all levels
Finish the Tutorial
Finish a level with 4 players
Bake 100 Pies
Visit the tent in Camp level 3
Dispose 50 diapers
Dispose 100 diapers
Have a baby equip a purchased blanket, toy, makeup and a hat
Avoid a projectile by turning into a ghost
Never miss a task from a baby (25 levels)
Put 25 snails in a terrarium
Thwart 50 kidnappers with diapers