Kaçmak has the sense of running away from something, while kurtulmak means to survive without (much) damage. You can "kurtulmak" from something while you "kaçmak". Sometimes, you can "Kaçmak" without getting "kurtulmak" in the end.
Don't I make a lot of sense today? ;p
As far as i read about it, could can be used for offers. So can't the sentence "You could escape with me." be correct?