With what you're describing, grab a brand new soft toothbrush and some clear dishwashing soap (Dawn is what I like). Do a warm water scrub of each ring with the soap and water, then dry thoroughly. Buy yourself a pack of these: https://www.amazon.com/YUANBOO-Polishing-Buffing-Accessories-Manicure/dp/B09132B371/ref=sr_1_9?dchild=1&keywords=nail+polishing+4000+grit&qid=1624499854&sr=8-9

Use the 1000 grit if you have any scratches, then the 4000 grit for super duper bling, or just go straight to the 4000 grit. It's infomercial level cleanup..."Iused to have trouble keeping my rings polished, but then I found MagiGold, the magical gold polishing tool inspired by nail technicians everywhere!"

Seriously, though. I'm a jeweler who haaaaates polishing to mirror finish and these things rock on gold. Moments to a high polish. Be sure to get that specific manufacturer/grit combo - I've tried other nail buffs with considerably lower bling results.