We use TradeGecko for all our picking and packing. We dispatch orders in our own delivery vans, on pallets and through a courier. Although there are other warehouse management applications which do this, and big integrations like Shopify to help us do this, TradeGecko almost does everything anyway. So it seems unlikely that we should need to invest in an enormous new system to improve accuracy.
I'd like to suggest a warehouse app. To count goods onto shipments, tick them off on a small screen, then submit them so that they are packed as a shipement on the sales order. And then to select packed shipments, and mark them shipped.
At the moment we print off packing slips, and tick them with a pen, then sit down at a desk to mark them shipped. It's clunky and very 20th century.