Will picking a package on a product overwrite the product's weight?

No. The package only fills in the weight when the product's weight field is empty.

When you choose a package in the Package field on a product:

• Dimensions are always copied from the package, so the parcel size matches the box.
• Weight is filled from the package only if you haven't typed a weight yet. The package weight is meant as the empty box weight, so an existing weight is kept.

Check the numbers and click Save on the product. To set package sizes, go to Settings → Store Settings → Shipping Package.