Actually the stolon look pretty normal, they usually aren't green in the
root areas.
Walt made great suggestions for getting to know the quirks of your area, be sure to do those.
While you are waiting for the soil test results; be aware that St. Augustine grass is
water hungry, it will die without
enough water in the soil, get a bottle of fish emulsion and use it to water your lawn with, the nutrients are not only organic but your grass tends to crave those nutrients as do your soil microorganisms.
As Walt said, a talk with your extension office will go far towards your grass education as well as any other items you want to grow, they are the
local experts, use them all you can.
Redhawk