The Liquid Plum'r has worked worked for now, but the clog will definitely be back in a few weeks. Fresh water is draining and somewhat quickly. However, if the pipes were complete clean, I would think the fresh water would drain more rapidly.
I looked at the plastic hair/gunk removers on Amazon, but, in the reviews, some folks have had them break inside the pipes. I am semi-handy with tools, but I think the odds are that I would do more damage with something like this. Plus the longest plastic ones that I have appear to be around 18" max. So they work for sinks, but probably not for tubs/drains.
The hand crank auger I ordered goes up to 25 feet, and I figure I can use for the other sinks in our house, as well as bathtub/shower drains.
We'll see. Hopefully, it will work. I've read that if there is a blockage in the pipes/P-trap, it will be disgusting when I (hopefully) pull it out. But it will make a huge difference. I hope this is it, as it means that was probably the issue, and that the Liquid Plumb'r was making enough of a passage to let the standing water through but not fixing the main problem.
If this is successful, as a preventative measure going forward, I'll use the baking powder/vinegar combination folks here have mentioned, which I think is far better for everyone and everything than Liquid Plumb'r or Drano. That, plus monthly plunging and pouring boiling water, will hopefully keep things in good running order.
If all of this doesn't work, I'll take the trap off and start augering from there if I can't do it from the drain on the sink itself.
Thanks again to everyone for the great ideas for this nettlesome issue.