Launcher login screen is double typing my password to programs running in the back!

[Enkidoh]

Welcome to Eorzea!

That sounds very strange - assuming it's not a setting in Windows for accessability that is set to copy your key presses, it's possible this may instead be a nasty piece of malware has ended up on your computer called a keylogger that has inadvertently glitched and is merely copying your password into random things instead of recording it and sending it to someone else to steal. A way to protect your password is instead of typing it in with your actual keyboard, is to use the 'software keyboard' provided on the launcher beside the login section - this is a grey button beside the login fields with what looks like a keypad on it.

The software keyboard requires you to use your mouse to press the onscreen key buttons to enter your password, which is invisible to malware like keyloggers and thus would stop your pasword being stolen (I've used it for logging in on both FFXIV and FFXI since SE first introduced it years ago and I've never had any problems with it).

Running antivirus and malware detection software like Malwarebytes might help.