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As used in eastern Carteret County in NC, it is an interjection indicating extreme doubt or incredulous disbelief.....due to the Down East dialect, it is often pronounced "droim"....(note: it is NOT an expression of dismay as another poster has stated)
You caught 50 boxes of flounder last week? Drime, I reckon you did!

For an EXTREME expression of disbelief, it is sometimes followed with "down"
by BigRedDog126 April 17, 2011
An expression of absolute dismay. Similar is usage to damn with no redeeming qualities. The quantative statement of one having been screwed, blued, and tatooed...

From Old English, Down East dialect
(looking down the barrel, .5 seconds before the lights go out) "Drime"...
by Barely Human September 26, 2006