Is your grass growing flat to the ground or more upright? If flat, it could be crab grass. Crab grass is an annual, but the plant makes gazillions of seeds that are happy to germinate once the soil warms up. If it is more upright, then possibly nimblewill which is a perennial grass. Both plants are warm season grasses (meaning they grow like gangbusters after it gets warmer.
The smothering techniques with cardboard/mulch should work (probably better on the crabgrass since the nimblewill might creep out from underneath before it croaks). A long term strategy is to make sure your natural area has more cool season sedges. These will shade & cool the soil and take up space before the the opportunistic warm season grasses take over.
