There seem to be different explanations to the German surname Dröge (Droege):
North German (Dröge): from Middle Low German droge ‘dry’, probably applied as a topographic name for someone who lived in an area of dry ground.
(
http://www.ancestry.com/search/SurnamePage.aspx?sourcecode=13304&html=b&fn=&ln=droege&submit.x=0&submit.y=0)
My book on German surnames (Rosa und Volker Kohlheim, Familiennamen, Mannheim 2000) says, it's from a byname "dürr" (meagre).
So "dry" would be the original meaningin both cases, and this seems to be the root of the word "drug" as well.
But once more my question: Where did you find, that DROGO was an old Germanic name? I've never come across it.