If I understand your question, the answer is no. 

Catanese produces examples of surfaces with $K_{Y_0}$ big and nef but not ample and such that if $Y_0\to X_0$ is the canonical model, then $X_0$ admits only RDP singularities which do not admit any global smoothing. 
Now if $Y\to \Delta$ is a smooth family such that $K_{Y_t}$ is ample for the general $t$, then I guess what you are asking is not possible.