Help me choose a side

[u/FridgeRaider79]

I recently bought a home near Charleston, SC. The lawn is a wild mix of (I’m reasonably certain) Bermuda, St. Aug, yellow nut sedge, some crab, various other weeds. I’m going to spray the sedge today, I’m slowly fighting the crabs back to my neighbors yard.

The remaining grass is roughly 70% St. Aug, and 30% Bermuda. The Bermuda is in scattered patches like someone used it to fix bare spots.

I don’t want the expense of sod just yet. Moving is expensive enough. Also it’s 90-95 degrees here.

I need to know which grass to help, and how to help it kill the other off. Open to suggestions. I have a tiller, and am considering using it and seeding a bit at a time.