My Widget Coding Disappears But Remains on Page

Our programmer created a custom theme for a site that needs a homepage cleanup. Then he left our company and I am trying to finish building. In the widgets area is a place for the homepage column content on the bottom. When I put anything in the widget and customize then it hit save it disappears but shows up on the page. Furthermore, if I update the neighboring widget it will frequently delete the code in the other widget.

Can anyone tell me where the code is going? Or how to make my backend widget display the content?

I have no idea how to edit it.

My staging site is located here: