Thanks for the edit/ update!
Honestly, walkability isn't a strong point in a lot of communities in Coastal NC, nor is public transportation for that matter. With that said though, it's not entirely impossible.
Since budget isn't an issue, but walking access to amenities is, I would first look at
Mayfaire Town View located in Wilmington's Mayfaire Town Center. They're actually Condos but I've noticed listings in the past for rentals in the complex (so you'd probably be renting through a private LL/ private rental company as opposed to an apartment complex per se- from previous rental listings I've noticed they rent from about $1200+ a month). You'll be steps away from groceries (Fresh Market), a movie theater, cafes & restaurants and shops. I honestly don't know what the level of safety is like in the apartments, but I've never felt 'unsafe' at Mayfaire Town Center by any stretch of the imagination- it's very well kept up and maintained so I would be surprised if they had any significant problems.
Additionally, the WAVE (Wilmington's public transport bus) runs directly in front of Mayfaire Town Center, allowing easy links to Downtown Wilmington, etc.
If I had your requirements, and no budget limitation- they would probably be the ones I'd look at first! Wilmington would be my first choice in Coastal NC regardless though, purely due to the fact that it's the largest city on the NC coast so has more amenities than the rest. With that said though, the downside is that unemployment in NC, and Wilmington is high (amongst the highest in the U.S) so I would strongly suggest securing employment first!
Good luck to you and your family! I hope that helps! I look forward to seeing what others suggest also