I think I found the solution to my issue, I am not sure if this will be of help to others.
The plugin I had for Table Rate Shipping for some reason was pointing to the Plugin called “Table Rate Shipping Module” by Nethues India, which has not been updated in 5 years! I noticed this when I clicked “View details” on the plugin.
That was strange though because I was sure I had a subscription with Woocommerce and on the Woocommerce page for their plugin it shows that it was last updated in May!
I therefore deleted the plugin from my installation and I downloaded again the .zip plugin file from the official Woocommerce site. I reinstalled and now this seems to allow me to change the settings with no issues.