I am a current resident of League City and the South Shore area. Overall traffic isn't too bad. It flows and isn't too crowded during off peak times.
Rush Hour is a different story. It is more congested. The "forked" intersection at FM 270, FM 518 and FM 2094 (Marina Bay Drive) is a pain in the morning between 6:30 am and 9:30 am. Then again around 4:30 pm and 7:30 pm.
You manage to get through it, but if you're heading south during those times, take South Shore Blvd over to FM 96 (League City Parkway). If you're heading north, stay in the right lane once you get to the light at Davis Road across from Clear Creek Intermediate School on your left, if you are leaving home at those times.
A few trips driving around town, you will find out which route is better for you. Local business like Walgreens, CVS, etc. close their doors around 10 pm. Traffic is very light then. However, people can be frisky on the road on Wednesdays through Saturdays because the local bars and restaurants become active.
Other than what I just mentioned, driving League City and the surrounding areas are relatively easy compared to Houston and other suburban areas around Houston.