Your bike looks great to me. Most of my rando friends ride bikes set up just like that. RF has good suggestions. Seconding what other people have said, I'd rotate your present bars CCW as seen in the photo, until the bar top behind the hoods is almost level. You should be able to put your forearms horizontal and have your wrists rest comfortably on the bar tops while you hold the hoods. You may find this position faster than using the drops, as the wind sees less of the cylinder of your arms.

Once you do that, a very comfortable grip is to put your thumb over the top of the brifter, 2 fingers above the lever and 2 below. You have to move your hand slightly to operate the brakes, but it's fine for all non-paceline activities. I use that position more than any other.

The other thing you can do is situps and hyperextensions, especially in the fall, and winter. Hey, you're doing great!