Is there a way to show or filter the sales order lines that are partially or fully paid already on a Transaction Saved Search with Type = Sales Order?

not sure about paid, but you can filter out the lines which are invoiced.

Unfortunately, I need the paid ones. I tried the Billing Transaction : Amount Paid field but it's not working when type = Sales Order.

You can achieve this using a custom column field and a userevent to update the field when the payment is received against the invoice.

Here's what I'm thinking. I'm not actually a script expert. I have a custom transaction body field on the Invoice. I want to update its value to Partially Paid or Fully Paid when a Customer Payment is applied against it. I'll use a user event script for this one.

Isn't that already happening? Invoice status changes to "Paid" if it is fully-paid. Otherwise, it remains Open

If it is PARTIALLY paid, it still says Open.

Why don’t you make it easier on yourself and do this search on invoices instead of sales orders?

I need to get the Item Fulfilment COGS which cannot be retrieved if the type is Invoice. I also need to get something from the PO if it is a drop ship which is only possible if type is SO.