Strictly speaking, the masses adopting rhotic pronunciation was, originally, an effort at mimicking a privileged-class affectation. The extra "R" in "drawRing" did not mimic anything, though, but seemed required by the imperfectly deduced rule. Those in actual contact with the privileged class had plenty of examples to, er*, draw upon, denied to those without.
Class markers in speech are always a moving target, a sort of low-grade arms race: low stakes for the upper class, high stakes for lower.
So, pronouncing it "drawRing" is a lower-class marker, similar in a way to "aks" for "ask" in American English.
As always, all of these pronunciations are legitimate and produce no misunderstanding.
[*] "er" is pronounced "uh" or "eh", rhotically.