I think I came up with a possible fix. I remembered that Safari has the option of opening any link into a new tab. I had mine set to Never. The other choices are Automatically and Always. I set mine to Automatically and now the tabs are behaving properly. I haven't had trouble closing any yet. Seems to make sense that the Never setting would interfere.
Not sure if it made any difference, but I stepped out of full-screen mode, set the preferences for the tabs to Automatically, and then returned to full-screen mode.