[tony_sologubov]

What you need to do is:
1) Create 'Product tax class' and assign it to all products, while shipping methods will still have 'Default tax class' assigned
2) Your tax rates must be applied to 'Product tax class', not to 'Any class' or 'Default tax class'.

You can create new tax class in the Store setup > Taxes > Sales tax tab
You can assign this tax class to products via import or on products details page.

Please, let me know if it helps.