Thanks for all the insight, folks.
I became so frustrated, I nuked my install and installed the latest ubuntu, fedora, and manjaro versions; I played with them for an hour each and suffered no dns issues. I then reinstalled solus and experienced the problem within ten minutes. In all three of those other distros, my /etc/resolv.conf contained the localhost--i.e., 127.0.0.1--and not the ip address of my router--i.e., 192.168.1.1; however, solus is different. After reinstalling ubuntu to troubleshoot some more, thinking I could live with going back to that OS, I found out I couldn't...
I've grown very fond of solus, particularly the boot speed; so, I'm back. This time, one of the updates available after install was for NetworkManager, which is what I believed was the root of many issues. I hope--fingers crossed--my issues are resolved and will post back...
Again, thank you!
EDIT
Well, long story short, the dns issue did not go away. After a reboot of solus, it's back. I don't know what the issue could be, but it renders solus unusable.