The expectation of the sum of two random variables is the sum of their expectations.
$$\mathsf E(X_1+X_2)=\mathsf E(X_1)+\mathsf E(X_2)$$
This is called the Linearity of Expectation. It works whether the random variables are independent or not.
It is a very useful thing to know.
So whether drawing with or without replacement the expected value is the same. $\tfrac {11}2+\tfrac {11}2=11$.
From first principles:
$$\begin{align}\mathsf E(X_1+X_2) =&~\begin{cases}\sum_{i=1}^{10}\Big(\tfrac i{10}+\tfrac 1{100}\sum_{j=1}^{10}j\Big) & : \text{with replacement}\\\sum_{i=1}^{10}\Big(\tfrac i{10}+\tfrac 1{90}\mathop{\cdot\sum_{j=1}^{10}}\limits_{j\neq i} j\Big) & : \text{without replacement}\end{cases}
\\[1ex]=&~\begin{cases}\tfrac 1{10}\sum_{i=1}^{10}\Big(i+\tfrac 1{10}\frac{110}{2}\Big) & : \text{with replacement}\\\tfrac 1{10}\sum_{i=1}^{10}\Big(i+\tfrac 19(\frac{110}{2}-i)\Big) & : \text{without replacement}\end{cases}
\\[1ex]=&~\begin{cases}\tfrac 1{10}\Big(\frac{110}{2}+\frac{110}{2}\Big) & : \text{with replacement}\\\tfrac 1{10}\Big(\tfrac{8}{9}\frac{110}{2}+\tfrac {10}9(\frac{110}{2})\Big) & : \text{without replacement}\end{cases}
\\[1ex]=&~\begin{cases}11 & : \text{with replacement}\\11 & : \text{without replacement}\end{cases}
\end{align}$$