Adding a 'Delivery Type' select field to shopping cart

I am trying to add a select field to the cart.php page so the user can select: collect from shop, local delivery or international. I am using Paypal Express so the user goes straight to Paypal once they click 'checkout' on the cart.php page so they don't add their shipping address within the cart. I have added a select field 'DeliveryType' to the page (just above the discount summery row). I have also added a column to the ecart object (I presume that is correct). I am stuck as to what to do from here.
I have created some shipping rules - dependent on the Session variable DeliveryType, but these don't seem to be working.

How do I get the shopping cart to update once a selection has been made? Is there a way to update the shopping cart automatically once the user selects a value?

I am obviously missing something. I have seen similar issues within this forum but none that directly help me.

I would just save the Delivery Type into a session variable.... not as a cart column since it shouldn't change row by row.

Then you can create shipping rules based on the session variable once it is created properly.

Thank you, I have got that to work. I need to record this SESSION value (DeliveryType) in the orders table after the payment has been take using Paypal Express. How do I do that?

Also, I have tested the order using a Paypal sandbox account and it all works except the shipping details. The user doesn't enter their shipping details in the checkout as I am using Paypal Express - so the user enters their details directly to Paypal. However these aren't being transferred to the order table after purchase. I have attached some of my files (I don't know if you need them)

Update the server behavior on the pp_confirm.php page named "Store Cart Summary In Database".

You can add any columns you need to the Orders table in your database and use that server behavior to record any information including session variable values that you would need.