If public schools are a requirement, St. Johns County is the better bet. Lots of new construction and the trade off for the schools will be a commute and cookie-cutter neighborhoods - not Houston bad commute, but you'll see traffic.
There are some really good public schools in Duval (including one of the highest rated public high schools in the Country - but it's not in a good area). The rub is there is no local path that will keep them in good schools all the way through K-12. At some point, you'll have to enter the lottery system and commute to the magnet schools. Many people go local public magnets through elementary then move to private schools for jr/sr high. There are several very good private schools in Duval (Bolles K-12, Episcopal K-12, Bishop Kenny 9-12, among others). Not cheap, but Florida will give about $8k/yr towards tuition to private schools under the expanded voucher program.
Northwest Jax is a nogo, same with most of downtown. Agree that west jax isn't where you want to be either, but Ortega / Riverside is west of the river and very much worth considering. Also an relatively easy drive to the Corps building. San Marco is another good option near the Corps and several good private schools (and one good magnet elementary & jr high). Home prices will be higher in Ortega and San Marco, accordingly and the homes are older (historic neighborhoods). Mandarin is also worth considering in Duval, south of downtown and some nice neighborhoods.
South of Jax (Flemming Island, Oakleaf, etc.) have some nice neighborhoods with good amenities. Commute would be rough and not worth it, IMO. But, people do it. Probably more house for the money down there.
We chose the intracoastal west area (Hodges and Beach Blvd, for reference) due to proximity to beaches and town. You'd be about 20-25 min to Corps offices, depending on specifics.
If you want to live at the beaches, I'd recommend Atlantic Beach or South Jax Beach. Central Jax Beach isn't as safe. Neptune Beach is generally fine. There's Ponte Vedra Beach in St Johns County, very nice (Sawgrass CC & TPC) w/ good schools. Commute from any of the beaches will be pretty far - PV and AB will be longest - but when you're home, you're at the beach. Note that homeowners insurance will get higher east of the ICWW.
Good luck and welcome to Jax. Happy to answer any questions.