Gene frequency problem
I think you pretty much nailed the problem. If I understand correctly, you are just confused about the approximation that your textbook does, an approximation that is correct but not needed. Let me restate the solution to the problem first.

Resolution of the problem

As you said the frequency of sick people $\frac{1}{300} = 2pq + q^2$ in this case, where $q = 1-p$ is the frequency of the (dominant) allele causing the disease. It is a simple quadratic equation with one unknown so one doesn't even need to make any approximation to solve for $q$ (or for $p$). Rearranging the equation and expressing $p$ in terms of $q$ (p=1-q), you'll find that $q ≈ 0.00167$.

Are the results intuitive?

Note that the frequency needed to create a frequency of $\frac{1}{300}$ sick individuals is $\sqrt{\frac{1}{300}} ≈ 0.05$ if the allele causing the disease is recessive while it is only about $0.0016$ (31 times less frequent) if the allele causing the disease is dominant. It shows that indeed a lower frequency of a more dominant allele is enough to create more sick people.

Is the $q^2$ term negligible?

Given that $q ≈ 0.00167$, the term $q^2 ≈ 10^{-6}$, while the term $2pq ≈ 10^{-3}$. So, indeed the term $q^2$ is rather negligible compared to $2pq$.