"It still appears on new invoices after the terms on all customers are changed"

Assigning customers a new Default terms, means that is used when you Create Invoice for that name.

"and changing the terms right on the invoices doesn't make it go away."

Make "what" go away?

Terms is how you help the program know to set the Due Date based on the invoice date; you also can select Terms on that invoice, such as when you Forgot to set up the customer terms or this specific sale falls under Different terms. You also can simply set the Due Date specifically, such as, when Terms doesn't apply to this sale at all.

"Past Due" doesn't show until/unless something is Past Due. If you are creating Historic Invoices, such as catching up year end 2017, and have yet to enter Payments, then it might be past due, because you haven't done all the tasks, yet.