Can I show different delivery dates by zip code, state, region or city? (Essential Estimated Delivery Date app)

No. Widgets can be targeted by Shopify Market only, not by zip/postal code, state, region or city. Shoppers can't enter a postcode to get an estimate either.

You choose which countries belong to each market in Shopify → Settings → Markets. Then you can create one widget per market with its own processing and transit times. See How to set up Estimated Delivery Widget in Different Markets.

If your delivery times vary within one country, set the transit range wide enough to cover all areas (for example 2–6 days). You can also mention in the description that delivery times vary by location.