Nick here from Shopify.
With something like this, it is difficult to find out what exactly is going on without seeing your shipping settings. From experience, mixing weight and cost shipping can be tricky at times, and I would suggest sticking to either one or the other. Do you think there is a way you can edit the shipping settings to stick with one way or another here? I also just want to confirm something with you so we're on the same page, do you mean you have free shipping for when it is one product, but if there is a second product added you would like there to be a shipping cost?
Should you want someone to look more deeply into this and why it could be happening, it could be worth reaching out to our support team directly as their phone and chat interactions are live.
Looking forward to hearing from you.
All the best, Nick