If there is any information that I can provide to help make this a better question or to assist in getting a better answer please let me know in the comments and I will update.
Please check out my photo below (sorry about the quality), you can see that I've removed the trap (elbow) and I have the snake in the outbound pipe. The snake makes it to the wall (about the length of the pipe) and comes back with black gunk but can go no further. It's my first time in the kitchen area with a snake so I'm unsure what to make of this...
I guess I should try turning the snake downward somehow but before I start really mashing about down there, I've stopped (since there seems to be a hard wall or something inside). This seemed like a good point to reach out and ask someone for a bit of guidance since I don't want to make things worse.
Any thoughts on what I can do at this point or how to work that snake further down the pipe?
I have the issue where the kitchen sink is clogged when I turned on the garbage disposal unit, the water backed up into the two sink (about a 1/4 of the sink full). I tried the Hair and Grease drain cleaner (from Home Depot ~ the heavy duty one sold in the plastic bag) overnight and the water did drain out of the sinks. When I tried to run water through in the morning, the sinks began to backfill again.
Per request please note the photo of the outflow pipe below.
When I tried to rotate the snake clockwise I was able to get 7-8 feet further. But afterward (maybe an hour or so later) the water starts to backfill even more after I replaced the trap and did NOT run new water down (https://youtu.be/Mcxoekjoazk). Now clear water is coming up into the sink. In fact, the water in the kitchen sink is (several hours later) now rising to about 1/3 height of the sink.
I'm posting this update for the next person having this issue. I was able to use the auger on the outflow pipe (going into the wall and downward). Once I did so I hit a blockage that felt like a brick/cement/hard packed dirt. I then went out to Home Depot and picked up a medium sized bladder(1-1/2 to 3 inches). Attached that to a water hose, inserted it up to the first turn of the outflow pipe and turned on the water. I noticed that the water sound changes as it eliminated the blockage and became fully smooth running after 10-15 mins. I gave it quite a bit more time just to make sure and now the clog appears to be completely gone. Hope this helps anyone with the problem and thanks to Ed Beal for all the advice.