If an address fits two of my shipping zones, which rates does the customer get?

The most specific match wins. Wcart checks the address against every region in every zone and picks the closest one:

Pincode first, then City, then State, then Country.

So a Chennai customer gets your Chennai zone, and the rest of Tamil Nadu gets the state zone.

One thing to watch: if the best-matching zone has Status set to Disabled, Wcart does not fall back to the wider zone. That customer sees no zone shipping options.

To set up regions, go to Plugins → Shipping Zone, click Edit on a zone, and use + Add region. Leave a level as Any to cover all of it.