Hi Melissa, 

I am sorry you are dealing with this! 

First, to confirm your ID - bindweed looks like a morning glory type plant with arrow shaped leaves opposite each other along the stem. It spreads horizontally and lies close to the ground, with taproots that can reach several feet into the ground, making it hard to destroy. When the stem is broken it exudes a milky substance. It is a very aggressive invasive weed so eradication practices will need to be extensive and diligently performed to rid your land of it. 

You mention potatoes, which are particularly susceptible, so if you did plant them, ground cloth would help, and consistent pulling of any signs of bindweed emerging is also helpful. Mowing or tilling doesn't help as the rhizomes remain in the soil and will regrow. With the vigorous growth of bindweed, I would recommend (if pulling is your chosen course of action) pulling on a biweekly basis for several growing seasons to gain control of the spread. 

The piles of unused soil would likely require the same consistent control methods, so it depends if you would rather spend a lot of time controlling the weed or a large chunk of money for removal. 

The below resources have further information on bindweed and control methods both chemical, biological and cultural:

https://extension.oregonstate.edu/news/controlling-perennial-bindweed-takes-persistence

https://extension.sdstate.edu/field-bindweed-control-yards-and-gardens

https://adams.extension.colostate.edu/ag-acreage/field-bindweed/