I figured it out. The way to clear the UPS quote is to do the following:
In Dreamweaver go to WebAssist -> Cookies -> Clear Session Values
In the "Cookies: Clear Session Values" dialog box uncheck "Clear all session variables".
Next, scroll down to the specific session you are looking to clear. If you do not find the specific session variable then click the + button and type it in. In my case I typed in "WA_Shopping_UPS_Quote" but this session name will vary depending on what you called your shopping cart. Next, set the trigger to "Button: YOUR_CART_Clear_100_pressed". (YOUR_CART is the name of your shopping cart)
The next time you need to have the UPS shipping quote completely removed from the shopping cart, simply click the clear cart button and voilà! - The UPS quote is destroyed. I hope someone else finds this helpful as this information can potentially save you several hours of trial and error.