Can I say "Until it is the only thing driving you" or do I have to change it to "Until it BECOMES the only..."?

This is a matter of choice.  Both work.  They have slightly different ideas.  Becomes implies completion of the process of change, whereas is implies the existence of the change (which, in this case, can only happen when the process is completed).  different ways to look at the situation: ending of a process or its result becoming true when the process is completed. 

Both are correct. 

Yes, one could say the former.