Add 2 to all parts of the inequality.

$$-\frac{1}{2} < -3x - 2 \leq \frac{1}{2}$$ $$-\frac{1}{2} + 2 < -3x - 2 + 2 \leq \frac{1}{2} + 2$$ $$-\frac{1}{2} + \frac{4}{2} < -3x \leq \frac{1}{2} + \frac{4}{2}$$ $$\frac{3}{2} < -3x \leq \frac{5}{2}$$

Remember to reverse the inequality signs when dividing by a negative number.

$$\frac{\frac{3}{2}}{-3} > x \geq \frac{\frac{5}{2}}{-3}$$ $$-\frac{3}{6} > x \geq -\frac{5}{6}$$ $$-\frac{1}{2} > x \geq -\frac{5}{6}$$

The interval is $\left[-\frac{5}{6}, -\frac{1}{2}\right)$.

Comparing this with the given options, option D matches our result.

The final answer is $\boxed{\text{D}}$.