To trace the orders/invoices that hare paid (or unpaid) there are two choices:
The first one is to use the custom order flags feature. You can rename them as you wish to show for example "paid" orders/invoices and then tick the flag of the appropriate invoices to flag them as paid or unpaid (according to your requirements). You may also create a custom view in your dashboard to display only the unpaid (for example) invoices/orders.
The other one is to integrate an accounting solution like QuickBooks Online (Megaventory integrates seamlessly with it) and have full management of your payments.