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After doing some research, you should be able to use any “member airline lounge” when you fly a Star Alliance flight while holding the Star Alliance Gold level as you say you hold.
This means that, lounges like Plaza Premium or Al Majlis at DXB will not be allowing you in for free (you can pay to enter those) but you should be able to access any airline member lounge so your only option to access using your Star Alliance Gold membership would be the Lufthansa lounge (they founded Star Alliance after all).
Similarly in Delhi international airport: only terminal 3 has international flights and there only the Air India lounge (part of Star Alliance) should allow you in with your Star Alliance Gold membership.
Generally speaking though, if you fly on business class or first class then that’s a different story because you would then be able to just go to the lounge reserved for business class ticket holders or the one for first class ticket holders depending on your ticket.
Hope this answers your question and have a safe trip wherever you go!
Reference link for Star Alliance Gold lounge access
DXB Lufthansa lounge details, airport website
Delhi Air India lounge details, independent review
DXB Lufthansa business class lounge / Gold members, independent review