Is there a way to have a "amount" field versus having to select a product?

Is there a way to have a "amount" field versus having to select a product where a customer can enter in a custom dollar amount to make a payment? This would be for a possible unpaid invoice etc.

Yes, when you set up your payment field, you can select the payment type as "User-defined amount":

Do try this out and let us know if that helps.