Dubrovnik - lodging neighborhood recommendations for 6 nights

Given the option, I would always stay in the old town. Evening is actually when you want to be there. That’s when things cool off and a good chunk of the tourist crowd leaves. Yes, it’s still crowded, but the town is basically wholly geared towards serving tourists and is very enjoyable.
And I’d much rather come back to that easily, rather than hopping a bus to Lapad or hiking up the hills after a day of sightseeing. I believe most of the island excursions depart from the Old Port too, which means you’ll have to make your way through town with all that entails if you want to do something like that. In my personas experience, just having the ability to take a quick breather in the old town in my own accommodation was worth the premium.

And let’s be real, you’re definitely going to pay a premium for this in high season. You will find there’s already limited options if you’re going this year. Whatever you decide, you should act quickly.

We visit Dubrovnik very often and always stay inside the wall. There is much to do in the evening, especially The Dubrovnik Summer Festival which will be held July 10- Aug 25 this year. No need to purchase tickets in advance.
This is one reason to get your accommodations ASAP.
There is kayaking around the wall, there is hiking on Lokrum island (they did raise the price of the ferry tickets quite a lot, around 25E) so spend the day there. You can swim, hike up to the fort, and have a lovely lunch at the only cafe there. Also, there is an old monastery with a GofThrones exhibition is you watched the show.
We sometimes go swimming off the pier, follow the walkway all the way around, in the early evening or after dinner.
A very enterprising, goth styled, young woman started Haunted Dubrovnik walking tours. We actually took two of her tours. She takes you all around the back streets of the city. We find these kind of tours full of history and a way to spend an evening out.
There is an evening cruise on an replicated Galleon with or without dinner. It cruises around Dubrovnik at sunset.
When you take the cable car up to Mt. Srdj, if you want to eat at the restaurant, I suggest you make a reservation. The views are amazing. but be warned, it is expensive, NY prices.

Like in any city, if you dig deep enough, there is plenty to keep you busy. Just keep an open mind.

Amazing advice from you both, thank you! We will book in old town, and quickly. Our "main" trip is to Austria, we sort of tacked Croatia on to the beginning for fun, since flights were similarly priced as a roundtrip to Vienna. All of that to say, I appreciate the day trip ideas! And we just booked a dinner reservation at the Panorama Restaurant - was amazed at how booked it already is, which means I will get on top of booking everything ASAP.

Panorama was a real highlight, well worth the treat. Nice work! (And honestly, for as much as folks seem to talk about the price, I thought it was pretty reasonable.)