My phone's screen died but it still works. How can I get the text messages from it? People are still sending me texts even though I can't read them. I've gotta get a replacement quick, but I also need to get the important texts I missed. Is there any way I can get them through my pc?

To be clear, the screen is completely blank but I can still unlock it via touch pattern.  It takes a few tries but Android File Transfer on the mac will tell me if it can or can't connect.  If it can't, I just try again and I can get it.  So the touch part still works, it just doesn't show any information.

The phone is a Galaxy Nexus.  When I connect it to my Mac, I can see the files in it via the Android File Transfer window. However, I can't find the texts here.

I have the Android SDK installed and I was told I can do this:

adb pull /data/data/com.android.providers.telephony/databases/mmssms.db


However the device shows up as offline when I do adb devices.  I can't figure out a way get it showing up as online without being able to use the screen.
AndroidLost is an app that might fit your need. Though the aim of the app is different and can do much more, the abilty of the app to read sms and send it to email is an interesting 
solution to your problem.

Read its FAQ and Getting Started sections - you need not even touch your phone as long as it is connected to Internet for installing this app.