Every time I look at renting a car in Dubrovnik, the reviews are full of horror stories about hidden charges. Is there any way to actually avoid the drama?

Total rookie mistake is just picking the cheapest random supplier on a basic search engine. If you want to keep your sanity, use a solid aggregator like https://roscardubrovnik.com/ instead. It basically forces transparency on the local operators. Everything from the exact security deposit to the insurance details is spelled out upfront. You lock in the actual price online, meaning you can look the desk agent dead in the eye and say "no thanks" to all those ridiculous last-minute add-ons they try to scare you into buying.