As I understand it, the legislation is far more ham-fisted than simply "borking domestic DNS". More or less, it hands the government -- and private parties having sufficient clout -- the proverbial lead pipe with which to beat you.
Step 1: Block access. Step 2: Criminalize circumvention of those blocks.
Of course, enforcement may be selective. But consider who is doing the selection and under what criteria -- both present and future.
Anyway, most of the people in America and in the world do not have a single clue about changing dnsservers... Heck, most of the times they do not even imagine that a website domain relates to an IP address.
As an example, HTTP uses the "Host:" header which contains the domain name of the website in question. Deep packet inspection could work by recognizing HTTP communication, inspecting the "Host:" header and simply dropping traffic to censored websites.