Solution:

Let $S$ be the sum of all seven scores. Since all seven averages were integers, $S$ is divisible by $7$. Also the seven distinct scores are between $91$ and $100$, so $91+92+\cdots+97\le S\le 94+95+\cdots+100$.

Thus $658\le S\le 679$, and the possible multiples of $7$ are $658,665,672,679$. Since the seventh score is $95$, the first six scores sum to $S-95$, which must be divisible by $6$. This forces $S=665$.

The first six scores sum to $570$. The first five-score average was also an integer, so the sum of the first five scores is divisible by $5$. Therefore the sixth score is divisible by $5$. Since the seventh score is already $95$ and all scores are distinct, the sixth score is $100$. Thus, E is the correct answer.