See also:
They move because they are text not underlining.
Looking at your document, you are using the Enter key erratically. It is designed to end a paragraph, not a line.
Consider using Rich Text Content Controls inside table cells to type your information.
Create a Form Using Word Content Controls by Greg Maxey
Create forms that users complete or print in Word - Office Support
You want your Invoice Form to be a
template, not a document. The invoices will be documents created based on the template.