can you drive over the dam?
Only by using a boat can you drive over the dam!
The dam is posted off limits in a lot of places. Besides there isn't ever any water flowing over the spillway to be worth accessing below the dam.
There are some maps that contend country roads can get you near the dam base. Years ago I tried to drive north or east past the power station, that road had a "do not enter" and gate across what my particular map claimed was a county road.
There is a primitive camping area on the south west side. I have yet to see a human on shore using that area. I would think you could branch off the trail or road leading to it to come down to water front of the entire Little Brown Cove shoreline. This is pristine wilderness with plenty of biting critters, especially at the waters edge. Most of this shoreline is covered by lilly pads. The camping zone has a buoyed off swimming area in a small cove. It would give you access to a point with some grass around it, but I suspect it would be one H of a hike from the nearest road bed. The road leading to it starts at the park gate, and starts as a dirt road.
The whole north end of the main camping peninsula and north boat ramp is on can be partly or wholly accessed by walking the nature trail. A lot of that shoreline has thick reed standing between the trail and waters edge. Again plenty of biting critters, especially at the waters edge. The park is known to have rattlers, copperhead, and Water Moc. snakes, and scorpions protect yourself accordingly.
Otherwise speak some spanish or vietnamese and follow that mob.