Sure would be nice to have side view pictures of your handlebars.

Done. Will post more accurate pictures later.

go singlespeed

Yeaaaah, nope

You could switch your hand momentarily to the hood and shift from there. I assume you're not trying to shift while also maneuvering a tight hairpin and dialing it up to 400 watts all at the same time, so the little time it'll take shouldn't be an issue. Would also recommend being more proactive with your front shifting if you are not right now. As in, shift to the ring you'll be using before you get to the point where you need to make that shift, even if it also involves making a few rear shifts..

I also didn't like the brake lever being multipurpose and looked at the options from SRAM and Campy then I found a set of MicroSHIFT on eBay that looked like they might go cheap. I ended up getting them for something ~$30 shipped and gave them a try. I'm running 10 speed 5700 DR's and 6700 brakes, the Shimano stuff works great with the MicroSHIFT STI's. The MicroSHIFT lacks refinement, very industrial but gives good solid shifts with loud clicks to gives audio cues. I went to these from Claris 3x8 and much prefer the solid certain shifting of the MicroShift over the Claris. I do wish that MicroSHIFT made a set like this with rear exiting cables but their new series with that feature has the Campy type levers that stick out from the side, not like the 10 speed and below.

I founds these on a quick eBay search
Microshift 8 Speed Double Shifter Brake Lever Set Shimano Compatible | eBay
I would think you could find a set cheaper if you look and wait.

Maybe it's the angle of the pictures, but the handlebars and brifters look like they're tilted up to me. That'll make them more comfortable in the upper positions, but definitely more awkward to use from the drops. If shifting from the drops is a priority, you may need to compromise and level everything out.

They're actually lower than what they used to be when I posted this. It's better but still iffy

Yeah, looks like you moved the brifters up the bars a little? To me, it seems like you overshot the sweet spot -- the goal (generally) is to form a continuous line between the top of the bars and the hoods. From there, I'd rotate the bars forward so that the top part is level with the ground. It might not feel as nice from the top, but the drops should get a lot more comfy and usable.

It's your bike though! I always try to make sure my drop positions are comfortable and usable for long periods of time, and I encounter bikes in the wild from the opposite mentality -- the bars and brifters are rotated so far back that the brake levers are sticking straight out and there's no chance of using the drops, but whatever works for them!

I did that to try to alleviate some wrist soreness that I was having but have since then adjusted them to the position in the OP. I can't tell if it's still bothering me since, well, half of my body is sore after a minor fall that I had. Once I get better I will able to ride more and see if I can have better results when riding.

That sucks, heal up soon!

Either move the levers further down the drops or just tilt the whole bar downward a few mm.

I would do both.

I did the first one as much as I can, I think I will lose my "sweet spot" if I tilt the bars any further than what I have now.

Fair point. I understand the annoyance of millimetre tinkering

Downtube shifter