I have created an acoount on your demo site, and tried to create 2 different shipping address.
The problem is, your site using flat rate for shipping. So, when i change shipping address (yes, the saved address appear on the field), the shipping rate stays the same.
I’m not using flat rate, so the shipping rate is calculated from (i guess) field town/city or postcode. When i’m filling checkout field manually, the entry of town/city is populated AFTER i choose province. So, i’m not typing it manually. And the shipping rate is changed when i pick town/city.
Maybe that’s what causing the bug. Because when i make a 2nd address from my account dashboard, the town/city field is just simple text box, not some combobox which entry is populated from choosing certain province.