I use the macOS desktop app and it does indeed save to draft and have cross-posting controls. There are some things I prefer the web app for, such as uploading images and site management, and I don’t like that the editor window keeps moving every time I open a new one, but for me I just work around them. Give it a go and be sure to check the menus for various bits of functionality and if you have any questions just tag me and I’ll do what I can to help.
There’s also this in the docs: ℹ️ Keyboard shortcuts