Gold in the setting is the part that actually holds up. Diamonds usually get you almost nothing on resale, but the metal still has a melt value: weight times purity (karat divided by 24) times spot. A shop offer is lower than that because of their margin, but it gives you a floor before you accept a number.

If the piece is stamped, I check it with this scrap gold calculator before comparing buyer quotes: https://mygoldcalc.com/scrap-gold-calculator