I understand that the post-ganglionic neurones of the sympathetic system are adrenergic, but surely these neurones will be excited to the same extent as the parasympathetic post-ganglionic neurones (due to the ganglionic synapses being cholinergic), and thus target organs/tissues will be receiving equal and opposite sympathetic/parasympathetic stimulation, or rather equal and opposite activation of the muscarinic/adrenergic receptors. (Sub-question: Why does caffeine, a perhaps atypical acetylcholinesterase inhibitor have a greater sympathetic effect than parasympathetic effect?).
We were taught that ganglionic receptors have very high thresholds. So we need a very high concentration of the drug to produce the sympathetic effects. We even solved a hypothetical problem in class, where a patient presented with a pinpoint pupil (typical insecticide poisoning) and increased secretions but an increased heart rate (usually they present with bradycardia). We would still treat the patient with atropine since the combo of pinpoint pupil and increased secretions would be exclusively explained by insecticide poisoning (either organophosphate or carbamate). The increased heart rate could be explained by ganglionic action leading to sympathetic activation of the heart.
But as said this was only a hypothetical case and patients of organophosphate or carbamate poisoning always present with bradycardia. To elicit ganglionic responses the person has to consume a dose way higher which would certainly lead to sudden death.
Hope that helps :)
It's worth mentioning that, despite both neurones in your question being cholinergic, they aren't exactly the same: the pre-ganglionar neurones are nicotinic, while the parasympathetic post-ganglionar ones are muscarinic. And carbamates have more muscarinic activity than nicotinic activity.
PS: this is not a good answer because it simply leads us to another question: why do carbamates have more muscarinic activity than nicotinic activity? And the answer to this last question may be Polisetty's hypothesis.