If I was hit by another driver and my vehicle was totaled, can I sue the at-fault driver for the difference between the amount paid out on the claim and the amount owed on my car loan?
No, you cannot. The law does not care about what you paid or what you owe, because that is so variable: some people negotiate good deals, others don't; some people buy for cash, others finance (which increases costs); some get cars given to them (e.g. by family) or have trade-ins to reduce the price; etc. Rather, all the law gives you when your car is totalled is the then-current-value or "blue book" value of it, which is a consistent and fairly objectively derminable number. If you were paid the value of your car, that is all you are entitled to.