I've just replaced my computer and now need to reinstall Dreamweaver. I can't for the life of me find anywhere on Adobe's site to download it again. What steps do I take to download and reinstall software that I pay for monthly?

Download/install the desktop application, sign in, install the subscription application(s).

Creative Cloud for desktop
https://helpx.adobe.com/creative-cloud/help/creative-cloud-desktop.html

Sign out, Sign in | Creative Cloud desktop app
http://helpx.adobe.com/creative-cloud/kb/sign-in-out-creative-cloud-desktop-app.html

Activation & deactivation help
https://helpx.adobe.com/x-productkb/policy-pricing/activation-deactivation-products.html#activate-how-to

Install, update, or uninstall apps
http://helpx.adobe.com/creative-cloud/help/install-apps.html

Installing Creative Cloud Apps
http://tv.adobe.com/watch/cs6-creative-cloud-feature-tour-for-video/installing-desktop-apps-from-creative-cloud/