Willpower is a notoriously bad and unreliable strategy for accomplishing much of anything. If you find that your writing time is diminished by the distraction of the Internet, there are a number of programs that will lock down all of your computer except your writing software for a set period of time. I don't use any myself, so I can't make any recommendations, but I've had friends speak highly of
Ommwriter (I don't know if it's suitable for screenwriting or not).
The other good news is that Twitter will wait for you, too:
1. Reading Twitter
There are applications to convert Twitter feeds into RSS feeds, so you can plug them into an RSS reader and read them at your convenience (without losing them) rather than trying to keep up in real time.
2. Publishing on Twitter
There are applications where you can preload/preschedule tweets on a timer, so that you can maintain a steady broadcasting schedule without always having to be at the keyboard.
Any time you are a slave to your tools, you have the wrong tools or you're using them wrong.