The Flat rates you already set up would ignore any weight or packaging type settings, so you can configure each product with an accurate weight and package type to use for USPS rate lookups.
Go to Seller Admin>View/Edit/Delete Products:
Select an existing shippable product of yours and click Edit;
On the first screen, if Shipping/Buyer's Address is not checked there, tick that box;
Click Next until you reach the Shipping Options screen;
- Weight: [enter weight for one unit of the item, ready to box and ship];
- Package type: Regular Box;
- Capacity: 9999 [this will ensure that rate-lookups assume everything will be packed into a single parcel to ship together].
Click Next until you can Submit to save settings;
Repeat this for all your tangible goods;
When adding new shippable products, remember to enable Shipping/Buyer's Address.
Go to Seller Admin > Cart Shipping Settings and set up the following:
Shipping Origin: select your country and enter your zipcode (if you're outside the US, just enter any 5-digit number);
Shipping Destinations: if you will only ship to certain countries, select them in the list, or otherwise select nothing for Worldwide shipping;
Regular Box - Weight: [enter the empty weight of the box you use, including any padding/packaging material you'd normally use]
Leave all other settings blank or 0.00;
Click Next to save settings and proceed to the Shipping Rules screen.
In the Existing Shipping Rules menu, I will presume you've already set up your Domestic rules with Country set to your own country, so you'd just need to set up one more Int'l Rule with Country set to "All Countries" (i.e., "rest of the world", covering all Shipping Destination countries that don't have a Rule of their own).
If you will ship Internationally at a different rate:
Shipping Rate Calculator: USPS Priority Mail or USPS Cheapest (this would compare First Class vs. Priority and apply whichever rate is cheaper);
Country: All Countries (i.e. Rest of the World);
Leave all other conditions blank (or 0, same effect);
Click Submit to save the Rule.
Click Back to Admin, and you're done!