Google Earth shows the FM1375 to be more promising. At least there is a place to park;but it looks like a good walk to the point of land; the farthest out into the lake.
Yes, heading west you can park at the bottom of the 'hill' right before you cross the lake. There are a lot of spots on both sides if you walk the road. I would suggest to get there early (around sunrise) and you should have plenty of options. There are also trails that follow along the lake shore and lead to spots.