Hey Travis, thanks for following up!
Do you see the same issue when updating to version 4.3.5?
On the outset, this appears to be an AJAX issue where perhaps the AJAX method used by The Events Calendar might be conflicting with that of the theme or another plugin. The reason I suspect that is that I see this error in the console when inspecting your site:
Uncaught ReferenceError: debug is not defined at HTMLDocument.<anonymous> (tribe-events-ajax-calendar.min.js:1)
I do not get the same error on my test site in default WordPress conditions, meaning no other plugins installed and using the default WordPress Twenty Sixteen theme.
One thing you might consider is trying to disable AJAX, which you can do by navigating to Events > Settings and disabling this option:

The other route is to keep troubleshooting this further to see if you can replicate the same issue in a default WordPress environment. Or, spin up a new test site where the only installed plugin is The Events Calendar and the active theme is the WordPress Twenty Sixteen one.
Thanks!
Geoff