Thank you for your question. 

I would have to agree with you that this plant appears to be dichondra which is a very common type of turfgrass weed in St Augustine, I don't believe it to be a type of ginger or hexastylis plant.

I wish you the best of luck on your shade garden!

https://aggieturf.tamu.edu/turfgrass-weeds/

Hi, just an update - these have sent up a few flowers, which has allowed me to make an ID. I am fairly confident that this is Lycianthes repens, a nightshade native to Brazil.

Given that they are invasive, toxic, and spreading in my lawn, I figured the extension might want to know (perhaps they are not new arrivals, but I had never seen them before moving into this house). I will probably try 2,4-D on them once our weather stabilizes at an appropriate temperature. 

I believe I now have a better ID - Lycianthes asarifolia, aka gingerleaf. It appears on a list of known invasives published by the Houston Parks and Recreation department.

It is great to hear from you again. Do you have any images you could send of the weed with the flowers?

I was not able to find any more flowers in good condition, but I was able to find some spent flowers and buds. One of the colonies in my yard had a number of flower buds, so I will hold off mowing and check in a couple days.

Seeing the flower really does make all the difference! I would agree with you it looks like  Lycianthes asarifolia I will share this with other agents to be on the lookout for this particular weed.