Finding individual utility
Your mistake probably occurred at the final substitution stage. Optimal utility should be $2\sqrt{x^*y^*}$, which is not the same as $2\sqrt{x^*y_i}$.

In general, maximizing a two-good Cobb-Douglas utility function $u(x,y)=Ax^\alpha y^\beta$ subject to budget constraint $p_xx+p_yy=m$ has general solution 
\begin{equation}
x^*=\frac{\alpha}{\alpha+\beta}\frac{m}{p_1}
\quad\text{and}\quad 
y^*=\frac{\beta}{\alpha+\beta}\frac{m}{p_2}.
\end{equation}

In your case, the parameters $A,\alpha,\beta,p_x,p_y,m$ take the following values: 
\begin{equation}
A=2,\quad\alpha=\beta=\frac12,\quad p_x=1,\quad p_y=p,\quad m=1+py_i.
\end{equation}
Plugging in the values, you get optimal consumption:
\begin{equation}
x^*=\frac{1+py_i}{2},\quad y^*=\frac{1+py_i}{2p}.
\end{equation}
Hence optimal utility is 
\begin{equation}
u(x^*,y^*)=2\sqrt{\frac{1+py_i}{2}\cdot \frac{1+py_i}{2p}}=\frac{1+py_i}{\sqrt p}
\end{equation}