How can I get back the old split screen behavior from Mojave in Catalina?

There might be a better solution to enable the old functionality through the OS, but I would be surprised if so. But I've found the free, open-source application Amethyst has been great for my usage.

To make sure you're getting the exact functionality you want, I suggest going into Amethyst's Mouse Preferences and ensuring "resize windows using mouse" is enabled. Amethyst preferences