Certified GKH Gardening Expert
Unfortunately, your photo did not come through. I am unable to see the damage. From your description, it could be an infection.
These are pretty drought tolerant, so overwatering will be a concern. Many times, they will not need any supplemental water, unless there has been a drought.
I would recommend a fungicide, in case this is an infection, and to make sure that the soil is, very thoroughly, dried down to about 3 or 4 inches or so before thinking about watering them.
Here are some articles that will help:
https://www.gardeningknowhow.com/ornamental/shrubs/ligustrum/growing-ligustrum-shrubs.htm
https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm