St. Augustine will choke it out..... it's pretty simple water, feed, and mow.

I would definitely compost the yard to help moister stay in the ground longer, feed the grass, and and help the ecosystem under ground grow good creatures that the soil probably needs. I would be willing to bet the ground is pretty compacted.

Arguing whether, or not it's a weed is trivial.

I will respectfully disagree..... if you keep cutting the weed eventually the root will run out of energy, because the weed will have few leaves to produce, and send energy back to the root. It will die, and will not have had the opportunity produced seed and propagate.

Cut it and it keeps pulling nutrients out of the soil to grow back. It might die eventually but its not going to be fast.

As for seed, don't ever get in a fight with pigweed. Those bastards will be 4" tall and have a million seed on them already

I don’t get it though half the yard is great and the other is terrible. I do the same on both sides. Only difference is one side has tree so it gets more shade. Maybe I should water the side without shade more?

Damn......that one side looks great.

I would compost and water the otherside more. I threw 8 yards of compost on my yard, and it really helped my yard.

One other thing is the summer dont cut your grass too short. The sun can burn the roots if they dont get enough shade from the grass. Also just like the weeds the fewer grass blades the less energy getting back to the roots, and the faster the sun pulls moisture out of the ground. The neighbors on both sides of me cut their yards to short. Needless to say their yards look like crap.

This is what a difference mowing higher and composting can make. It's easy to see where our property lines meet.

Above is another good example, the healthiest thickest St Augustine will be 4" to 6" tall measured from the soil, not the runner / thatch / mat from mid summer to fall. Most of Tx from early to mid June to dormancy stage, then clip it down to 2.5" to 3" if you want it that short anytime before spring green up.

Spraying or applying granule herbicides without 100% knowing why, what & when does far more damage than good.

I’ll dig those Dallis up, water that one side more, and raise my cutting height on next cut. Thanks guys.

Everytime you mow, those little black seeds scatter and create hundreds of new plants.

The actual seeds from dallisgrass are not those little black specs, those are anthars.
That's where the fungus that can cause 'staggers' in livestock attaches / starts.

Dallisgrass seeds are flat roundish ovalish & range in color from green when immature to off white to yellowish to light brown when mature, and are much larger than the anthars, large/ mature plant can produce seeds almost the size of a bb.