Scottish dialect: soaked, drenched, or thoroughly wet; saturated with moisture or liquid.
From drouk (verb) + -an (Scots adjectival past participle suffix). The suffix -an marks the state resulting from the action of drouking.
A droukan cloth or person was completely soaked in old Scots—it's not just wet, it's the kind of drenched where water drips from everything, which happened frequently in rainy Scotland.
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