@blowthemakiss what you describe sounds more like scale insect. Mealy bug looks either like a white "mealy" bug, or a white fluffy mass of eggs. Scale insect has a white mass of fluff underneath when it's reproducing.
@667TheNeighbourOfTheBeast The yellowing of the leaves looks as though the avocado isn't getting enough nutrient, either because it's used up the nutrient in the soil and wants some fertiliser adding, or because the soil is too wet and the roots are rotting. The droopy leaves points more to over watering. This wouldn't be a result of you both watering on the same day, it would be a longer peiod of being over generous with the water.
By far the easiest way to get rid of the mealy bug on something with just a few big leaves is to pick them off., or dab them off with a paint brush.
If you can get the plant back to health, it will be far more successful at resisting mealy bug attack. Iget occasional mealy bug on my cacti, and it's the ones that are already having problems that succumb.
When you say "spraying with a mixture of neem oil and washing up liquid", do you mean just neem oil and washing up liquid? I wouldn't see that being very good for the plant. On its own, the neem oil I would be afraid of blocking the stomata which control water flow from the leaves, the washing liquid on its own would damage the cuticle of the leaves, mixed together they'd make and emulsion which I can't see as being good for the plant. Or do you mean water with a squirt of each in?