I fixed it by clicking the "Revert to Classic Template" button. Once I did that, I was able to go to the comments tab and reselect all of my preferences. (They were set to not allow comments, etc.) Before I reverted to the classic "Dots Dark" template, my comments preferences showed that comments were allowed, but somehow, that never got put into code.
Now, if I get an itching to customize my template, I have to pick the new template. If I go back to the "new and improved" version of this template, no comments.
I will have to figure out which is worth more: my link list and quotes, or the comments. In the meantime, I will probably start looking for another template that allows both. Any suggestions on where to look?