Help with reordering checkout page

Hi,
I need to move the payment options on the checkout page just after the billing details. I know I can do this with

remove_action('woocommerce_checkout_order_review','woocommerce_checkout_payment', 20 );
add_action( 'woocommerce_after_order_notes', 'woocommerce_checkout_payment', 5 );

but this also moves the terms and the “place order” button. I need to keep those two underneath the order review tabel though. How could I do this? Is there a function or if I had to modify the templates, where would I need to add the “form-row-place-order”-div?

Thank you very much!