1/2 right.You have 30 days to send in a address change. You current one will remain valid until the new one arrives.
You're old LTCH remains valid as long as you submit the change of address within 30 days. You will not automatically receive a new LTCH card.
If you want a LTCH that shows the new address, you have to pay for a duplicate card (there is an option for this on the address change form). The new LTCH will show the address change and have the word "DUPLICATE" over the signature line.