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.
I'm sorry but pretty much everyone who grew up with English as a second language would like to disagree.
Hearing "drawRing" to this day throws me off -- I have to repeat the noise to myself internally to make sure I understood it right.
Not to put too fine a point on it, but their opinion doesn't count.
Foreign speakers have no say in what's legitimate and may be confused by absolutely anything.
https://en.wikipedia.org/wiki/Hypercorrection
while thaumasiotes is referring to
https://en.wikipedia.org/wiki/Phonotactics
There are examples in the hypercorrection article about people adding an /h-/ in a very similar situation, so what you describe clearly can happen. But are you sure that the R example falls into that category?
It seems to be a distinction about how conscious the change is, or at what layer of language it happens. Isn't that an empirical question that could be tested?