Hey Tim,
That’s some odd behavior, can you share your system information with us so we can investigate this a bit more closely? Here’s an article that shows how to share that info with us: https://theeventscalendar.com/knowledgebase/sharing-sys-info/
In the meantime, you may also find it useful to leave things as-is on your site, and then simply activate a default theme like Twenty Fifteen on your site. Then, create a ticketed event with the 0 ticket cost and see if it works better when you try adding it to the cart on the front-end.
If not, leave this default theme active and then proceed with deactivating EVERY plugin on your site. Check on that ticket on the front-end and see if it adds to the cart successfully after each plugin deactivation. You should do this all the way through until The Events Calendar, The Events Calendar WooCommerce Tickets, and WooCommerce itself are the only active plugins left on your site.
This process may sound tedious but is fortunately quite quick to work through in actuality.
Let us know if you can do both things listed here and what you find – thank you!
George