To solve this problem, we first need to calculate the total number of different four-digit numbers that can be formed by rearranging the digits of $2023$.

Step 1: Calculate the total number of different arrangements.
- Since the number $2023$ has four digits with one digit repeating twice (the digit $2$), the total number of different arrangements can be calculated using the formula for permutations of a multiset: $\frac{4!}{2!} = \frac{24}{2} = 12$. However, this calculation includes permutations where $0$ is the leading digit, which are not considered four-digit numbers. The permutations with $0$ as the leading digit are $0232$, $0322$, and $0223$, totaling $3$. Therefore, the total number of different four-digit numbers is $12 - 3 = 9$.

Step 2: Identify the arrangements where the two $2$s are not adjacent.
- By examining the possible arrangements, we find that the numbers $2023$, $2320$, $2032$, $2302$, and $3202$ are the only ones where the two $2$s are not adjacent. This gives us a total of $5$ such arrangements.

Step 3: Calculate the probability.
- The probability that a randomly rearranged four-digit number (including the original number) has the two $2$s not adjacent is the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, the probability is $\frac{5}{9}$.

Final Answer: The probability that the resulting different four-digit number (including the original number) has the two $2$s not adjacent is $\boxed{\frac{5}{9}}$.

Therefore, the correct choice is $\boxed{A}$.