Not only do you need root, you also need to mount /system as R/W. I recommend the following steps, assuming you are already rooted:
1) Uninstall "Swype Installer" (optional; I did not have to).
2) Search for "mount /system (rw /ro)" in the Market and install it (free).
3) Search for "Terminal Emulator" in the Market and install it (free).
4) Open the mount app and click "Mount r/w". I do not recommend you check the box to mount as r/w at boot.
5) Open the terminal app and type the following ([Enter] represents the Enter key, so don't type it):
rm -r /system/lib/libSwypeCore.so
rm -r /system/app/Swype.apk
6) Reboot the phone (optional; I didn't have to).
7) Download Swype Installer (if you have it already; you don't have to redownload it) from https://beta.swype.com/android/get
directly on your phone browser.
8) Run Swype Installer and follow the prompts.
9) Go into settings, enable and configure your Swype keyboard.
Best of luck!
Those of you who switch between 2 languages often, you do not have to press and hold the Q letter anymore. Just swype your finger from the Swype key up to the Q key. The first time, you will need to select your second language. After that, every swype will toggle between the two.