The thing your build is most obviously missing is a bread-and-butter spell - something low-level that is usable with both Heart of the Storm and War Caster. You've ruled out thunderwave and shatter due to the noise, and while you didn't mention witch bolt I assume you've ruled it out on account of it being crap.
What you need is chromatic orb. It's 1st-level, on the sorcerer spell list, can deal lightning or thunder damage (handy for resistances or immunities), and works with War Caster.
After that, everything else is fairly minor. Since you only get 1/3 of the benefits of Sentinel, I'd recommend a different feat. Elemental Adept is a possibility, but if you're planning to be in melee, Crossbow Expert would prevent disadvantage on your eldritch blast and chromatic orb.
None of the Oaths really do more damage than the others as such. The only Oath feature that directly impacts your damage is the Oath of Devotion's level 20 ability, Holy Nimbus - it deals 10 radiant damage to any enemies within 60 feet of you. So, to answer your question directly, Oath of Devotion does the most damage.
This is, obviously, a fairly ingenuous answer. The real answer is that you can't look purely for damage.
For example, both the Devotion and Vengeance Oaths have features that increase the accuracy of your attacks, which in the long-term, will increase your damage.
On the other hand, the Oath of the Ancients has an ability that makes you and your allies much tankier and harder to kill. How does that boost damage? Well, you can't deal damage if you're dead, so anything that keeps you alive is also indirectly boosting your damage.
Then there's things like spells. The Oath of Vengeance gives you access to hunter's mark, which potentially lets you add 1d6 damage to every attack you make. But in practice, it's not quite so simple - it requires concentration, and you also have to move it around to make sure it's on the enemy you're attacking. And, obviously, it requires you to make a lot of attacks to maximize the damage it gives.
As a counterexample, the Oath of the Crown gives access to spirit guardians, which deals damage to enemies who come near you - under the right circumstances, this will vastly outdamage hunter's mark. But in the wrong circumstances, it might be completely useless.
So, in the end, there's no easy answer - you'll have to look at all the Oaths and determine which one you think will work best for you.
Best Answer
As expected, there is never one perfect spell for a given situation, especially since sometimes a spell list will have a similar but notably different version from others. I'm going to note a few spells that will be beneficial, sorted by area types.
Burst
Burst are possibly the easiest spells to hit a large number of enemies while minimizing friendly fire, making them a good choice even if some of the others are theoretically able to hit more targets. Fireball is also in this area, and the choice between it and sudden blight is debatable given the limited scope of sb's added effect, but sb comes online sooner and isn't necessarily as well known so I wanted to give it a shout-out. Sound burst doesn't have the area of other options, but its additional effects buys it a space on the list.
Cone
No list about maximizing area damage would be complete with the resident 60ft cone, the scope of which is tough to grasp in 3 dimensions. Elemental annihilation wave actually fits this question better, but costs 2 full turns to do so.
Cylinder
Oddly, the other cylinder I looked at (geyser) has a duration of 1 round and is therefore officially not compatible with Dangerous Sorcery. If your GM is willing to homerule it, geyser is a better spell in this circumstance for its larger area and additional effects.
Line
Lines. Nearly impossible to hit more than a couple of things, but theoretically so powerful. This one is hard to decide between
kamehamehainner radiance torrent and blazing fissure, so I went with both for different reasons. Similar to elemental annihilation wave, irt is probably the better answer but has issues due to it's abnormal casting.Other notes:
These spells might work with Dangerous Sorcery; they cause persistent damage but don't have a duration themselves. As a GM, I would rule against it (except grim tendrils initial damage), but both are good spells in their own right and would be excellent choices when using the feat, if allowed. Magic missile isn't helped enough to fully list it out, but suffice to say that you can hit 3, 6, 9, 12, or 15 different targets for 1d4+2 with the combo. Wooo.
Source: Used pf2easy.com/spellbook/ to filter to spells with the Traits: Acid, Attack, Cold, Death, Electricity, Fire, Force, Mental, Negative*, and/or Sonic. Also limited search to level 5 (some higher ones, such as prismatic spray or meteor swarm, would have been likely candidates), then sifted for area effects. *Added Negative under the pretense of the more common situation of facing living enemies; for basically any Negative spell there is a corollary Positive spell that I did not want to sift through but exists if you find yourself fighting undead.