Consider $X= Spec( \mathcal O_K)$ and an open subset $ U \subset X \quad (U\neq \emptyset, X)$.  
 Take two copies $U'\subset X',U''\subset X''$ of the above   and glue them along the identity  $U'\to U''$.    
You will obtain a scheme $\bar X$ that is covered by the two different open subschemes $X',X''$ each isomorphic to $\mathcal O_K$.    
 The scheme  $\bar X$ is integral, normal (since the open subschemes $X',X''$ which cover it are), it strictly contains two copies of $\mathcal O_K$ and of course is not affine since it is not separated. 

**Edit** When I wrote this answer I had started to suspect  that no separated example existed , but couldn't prove it. Fortunately Qing  has now provided  a (beautiful) proof and I'm very grateful to him (as is everybody else, I guess).