D&D kind of sucks, and especially sucks for what you want to do.

• It's not good for 1:1 games. It makes many assumptions about there being a full party
• It doesn't offer much advice for DMing, and has a lot of ways to fuck it up.
• It has like no rules for roleplay. It has very vague "If you roll a high number maybe they'll do you a favor" suggestions, but for something like "I want to stand up straight, look him in the eye, and tell him he should yield or suffer the wrath of my patron" it leaves it entirely up to the DM how to resolve. Other games provide a lot more guidance
• It is entirely combat focused. It is crunchy in annoying ways.

Chronicles of Darkness is modern day occult. it's not very light, but it's closer to what you want. I'm fond of Mage: The Awakening.

Fate is a good general purpose system that's more in line with how I think people imagine RPGs are. It's also free. It cares a lot more about the narrative than D&D. D&D puts most narrative detail squarely in "unimportant flavor". So if your character's story is they were a pirate, that doesn't do anything in the game unless you have the pirate background, and those tend to be very limited. Contrast with Fate that is much more explicit about "Aspects give you permission. If your character has 'Former Pirate' on their sheet, they definitely know some sea shanties and how to move around on a boat". D&D leaves that up to the DM. And then a lot of people are like, if it's leaving so much up to the DM why am I even buying this book.

Powered by the Apocalypse games are also popular. I don't personally like them that much, but I see why people do.