Yes - I tried tonyyoungblood’s drive cache workaround, and it appears to have solved my issue.
I had to run the autorun.sh file first (which I always had to do, because of the fact that it was not retaining the driver), and after that command was successful, I ran the update-initramfs -u command (specifically sudo update-initramfs -u), restarted the machine, opened a terminal session, ran lsmod, and got the r8168 driver in the list instead of r8169. From what I can tell with initial testing of large downloads and going to Internet web sites, it is functioning well.