Firstly, you're not the only one that can't change worlds. Most can now do so, but world changing has been an issue for nearly 2 months now, associated with login changes that have been made.
As you have clicked on 'change server' from the front page, logout of the game (the logout link at the top of the screen) then log in. Try clicking on play - whatever it suggests as the world, go there! Then, in options in-game try to use the in-game change route. That may or may not work, but you will not get a black screen. The standard advice also applies, clear cookies (related to EA and the game), cache and history related to the game.
Most have found a route into the world they wish to play - sometimes the work-around of a direct link is needed.
Here are links to the two worlds you've mentioned:
Tiberium 34 https://prodgame13.alliances.commandandconquer.com/392/index.aspx
Tiberium 35 https://prodgame08.alliances.commandandconquer.com/394/index.aspx.
