Update: I fixed it. I didn't know about what P.Philips mentions below, the feature concerning Optimized Battery Charging. The Apple Tech I spoke to had me running diagnostics and even suggested at one point that I get a new Mac as it did seem (their words) that "the ports were probably bad." Right.
Shame on you, Apple. I called you because I didn't know. If your tech support had me check that preference pane, I could have solved this sooner. Instead I get inexperience and bad advice. What a negative this company has turned into.