Don't worry, it's not just you. At the time of this writing, the option to start a game has been hidden by the Words With Friends developers. You can still start games, but you have to know what to look for - and it may not be apparent.
To begin a random game, tap the block from the Games tab that includes a random player's name. It's just no longer called random match.
The solo challenge is also hidden behind events now, which you can find on the Games and Events tabs.
If you want to start a game with a friend, you now have to go to your profile, open your friends list, and then select your friend to begin a new match.