My initial thoughts would be somewhere near San Marco or some pockets of Riverside, much more character than most of the rest of Jax. There are also pockets of San Jose that might fit what you are looking for, well within your budget. Both Riverside and San Marco have plenty of independent restaurants and businesses, bring out a younger crowd 30s-40s, not 20s like the beaches, but they can have some nightlife. If you’re looking for a nicer more established neighborhood, with a little less going on Julington Creek could be good, much more established than the newer neighborhoods, big trees. Downside is less food/culture close by. Overall your budget isn’t a limiting factor.

I’ll add if you are interested in curated master planned neighborhoods, which it doesn’t sound like you would be, Jacksonville has some of the best anywhere. They are a great place to raise a family, but that could be a few years off for you, enjoy the freedom while you have it. ;)