Hi Kyler!
In this case I would certainly advise that you take a look at the Better Shipping app, which allows you to add shipping rates based on zip/postal codes, so you can create a rate that reflects and compensates for how far away the delivery is:
https://apps.shopify.com/better-shipping
About your question about changing "shipping" to "delivery", you could change the name of your added shipping rate in settings > shipping to something like "Local Delivery" and this is what the customers will see on the checkout pages. In addition to this, you are able to change how shipping wording is displayed in your checkout by editing any shipping-related fields in online store > themes > edit language > "Checkout & System".
Hope this helps!
Andrew
Shopify Guru
support@shopify.com