Island expeditions don’t award gear, only AP. Good place to grind AP, but not so good if your focus is gearing. Warfronts are very solid, with the Arathi one awarding 340 per run and a 370 once per cycle on your first completion, and the Darkshore one award 385 from the first completion and either 340 or 355 per run (I don’t honestly remember). Warfront rewards can include both weapons and Azerite, though the Azerite items are exclusively of the Honorbound variety (so the generic trait is generally more oriented towards PvP).
World quests also scale up to 360 or so now, with certain high value ones (usually instance quests, like the Underrot one that was up the last few days (and might still be)) awarding 370. Invasions also reward up to 370 gear and can award both weapons and azerite, and often include WQs with high amounts of AP or solid gear rewards, too. Heroics and regular mythics are also solid options, though getting into regular mythics through the group finder can be a chore unless you have a really well-geared buddy to help boost you for invites.
Edit: Oh, and check your emissaries regularly! There was an emissary a couple days ago that awarded up to 370 weapons, and one that expired this morning or maybe tomorrow that awards up to 355 trinkets/rings.
Lastly, the AH has gear for bracers, gloves, belt, legs, and boots at 340 itemlevel that’s usually not too expensive, since they added it in 8.1. Crafted rings are still 310, unfortunately, and Azerite items cannot be crafted at all (excepting BoP pieces like the Engineering helms). On the trinket front, most of them are hideously expensive, but if you have the moola (or the RL moola to buy a WoW token to get in-game moola), a Darkmoon Deck: Fathoms is an amazing trinket. It will last you until you start getting 385 or even 400 trinkets, as it easily trumps every single other trinket in the game at 370 or below.
Edit: just noticed you’re a tank on the armory. If that’s your primary focus, skip the Fathoms, as it’s a DPS trinket. You can get a Darkmoon Deck: Blockades if you want for tanking. It’s not as insanely powerful as Fathoms is, but it’s still a very solid trinket, and because it’s not as grotesquely overpowered, it’s also typically massively cheaper than Fathoms.
As for PvP, I’ve heard it’s solid for gearing up, but I have no experience with how the gear progression system works for it in BfA. I’ll have to let someone more PvP-oriented answer that one.
Edit: oh, also, don’t enchant your weapons with the same enchant. You get much better mileage by using different enchants because of the way that the Navigation enchants work. For tanking, I recommend either haste/armor or versatility/armor (with either one being on either hand, it doesn’t matter). For DPS, it depends on your stat weights, but it’ll be some pair of crit, haste, or versatility. For a good hybrid set that’s solid for both tanking and DPS, go haste/versatility.