Thank you for your responses, everyone.
Uncle Turbo, The ignition switch is a possibility for sure. I agree that the symptoms are likely linked. The new distributor came from Rockauto. A friend who is a mechanic helped with the installation and I am confident (that between the two of us) that everything is wired correctly and I made sure that it is correctly matched to the car. The first coil actually died with the original distributor and I replaced the distributor as part of our attempts to fix that situation.
gsragtop, There is only one coil. The dielectric grease is a possibility. Not something that the repair manual instructed me to do with the installation--but still a possibility for sure. That is a good point about the resistor. I have also wondered about the quality of the replacement coils.
Tester, Great explanation. I have considered the control module and that may very well be the issue. The biggest problem for me if that is the issue is that is the part is $400.
Thanks again, everyone--for all of your ideas, explanations, and effort to help me with this situation.