Problem 209

Let's calculate the dot product of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$:

$$\overrightarrow{a} \cdot \overrightarrow{b} = (x-3) \cdot 1 + 2 \cdot 1 = x - 1$$

For the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ to be collinear with the same direction, we must have $\overrightarrow{b} = \lambda \overrightarrow{a}$, with $\lambda > 0$.

Setting up the equations:

$$\begin{cases} x - 3 = \lambda \\ 2 = \lambda \end{cases}$$

we find that $x = 5$. When $x = 5$, the condition $x > 1$ is satisfied, but in this case, the angle between the two vectors is $0^{\circ}$, so the angle being acute is not true.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ form an acute angle, then their dot product must be positive:

$$\overrightarrow{a} \cdot \overrightarrow{b} = x - 1 > 0$$

Thus, $x > 1$ must be true. This implies that " $x > 1$ " is a necessary but not sufficient condition for the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ to be acute.

Therefore, the correct answer is:
$$\boxed{ A }$$

This problem assesses the understanding of necessary and sufficient conditions along with the relationship between the angle of vectors and their dot product.