Partial receipt on purchase orders

Let's take this a bit at a time:

"We are using QB Premier"

You have to Install it as Retail, or as Wholesale/Manufacturing, to have the Function of Customer Backorder tracking.

"Once a Sales Order is created, if we create a partial shipment, QB is updating the quantity of the original Sales Order rather than leaving a balance open waiting to be shipped."

That's called Backorder Tracking.

"This is also a problem with Purchase Orders. If we create a Item Receipt for a partial quantity, QB is updating the quantity of the original Purchase Order rather than leaving a balance open waiting to be received."

For PO, you always have the tracking function, unless you set the Qty to 1. Example:

Engineering Contract of $15,000 is a PO for Qty 15,000 at $1 each, to handle Partial Payment Requests. And on the Purchase that you create, then select that yes, this is fulfillment of the PO, you always Confirm or Change the qty to match what you just got. That never changes the PO qty.

"How do we correct this?"

Did you install the right Premier Edition?