Apparently, pennywort loves my southern Mississippi grass because it is spreading like wildfire.
1. Will the vinegar solution kill my st. augustine grass?
2. What is the purpose of the molasses in the vinegar-killing formula?

I believe vinegar will kill your grass - or any other plant it touches. It would have to be used selectively. The molasses is probably included as a surfactant.

Is your soil dry? Real wet?
Do you have hydrocytle, Dichondra?
Or something else?

How short are you mowing this turf?

Did the vine come with the sod?
How shady is your yard?

I have difficult time caring about turf, but answers to the above questions might help someone else pin down a solution for you.

Try watering less.