Sounds like you may have scale insects as well as a fungal problem going on.* It sounds like you are watering too much if these are established plants.*
Btw, these plants are considered invasives in Texas, so if this is a new planting, I would recommend you remove them and plant something that isn't invasive.
Ligustrum also get white peach scale.
You can use an organic horticultural oil for the scale insects called Organocide.* It's made from sesame seeds.* Don't confuse it with the synthetic Organicide.* Note the different spelling.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.