This is how I ended up building the soffit. I decided to write this up as an answer, since there is not a lot of information, including on YouTube, dealing with building kitchen or basement soffits or bulkheads, where the joists are parallel to the planned soffit. Hopefully the readers can benefit.
Step 1. I cut back the existing ceiling drywall to the point where it wouldn’t interfere with the new soffit frame. I made sure that I didn’t cut beyond that, as you want your soffit drywall to butt under, and against the ceiling drywall, and not against the frame above. This way the soffit vertical drywall piece will support ceiling drywall from any possible sagging and will make this joint much less susceptible to cracks.
Step 2. I took the measurements and built the soffit frame on a level floor. First I built the front of the frame using a 2x4 as a top piece and a (ripped from 2x4) 2x2 as a bottom piece. I used the lumber as straight as I could find - a challenge in today’s breakneck lumber production, especially for boards 10’ and up. The ladder "steps" were made with 2x4 pieces at 24” at the center (you could go 16” but unnecessary for narrow soffit). I used long clamps like this, which made it MUCH easier to keep the ladder together while assembling than it otherwise would. I used 3" these decking screws to assemble the whole ladder.
Step 3. Next, I cut some 2x4s at a length that would fit between two joists and "toenail" them with 3” decking screws to the joists in a perpendicular fashion.
Step 4. Lifted the assembled front of the soffit, and attached it with the same decking screws to the 2x4 braces I toenailed to joists earlier, as well as the side walls.
Step 5. I attached the second piece of 2x2 from that ripped 2x4 with 2x4 ladder "steps" to make up the bottom of the soffit.
Step 6. Hang drywall as usual using drywall screws, tape the joints, install a corner bead (a lot of people use vinyl beads on soffits, I just prefer metal ones), mud, and finish.
I left the soffit on the back open (no drywall), as new kitchen cabinets, as well as the trim (still to be installed), would fully cover the opening. Had I had no cabinets, I would additionally screw the bottom of the soffit to the studs on the back wall and run the drywall all the way to the wall. In this instance it was unnecessary, and besides, I had a range hood vent hanging behind it.
Here are some photos of the build. I didn't take detailed photos of the whole process but it does show some detail.