If the only thing that changes is going from http to https, then you just need to change the checkout behavior on the cart page. Edit the redirect from checkout.php to checkout.php
if is is a shared certificate, then it will be a bit more involved, see the "Secure Socket Layer (SSL) and eCart sites" in the How To Section of the eCart support page:
I cant find any documentation on switching to a Shared SSL. Where could I find that?