No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically, see M. Stoll, E. F. Schaefer, How to do a p-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), 1209–1231, and the references therein. So if you knew finiteness of $p$-primary parts of sha outside a finite set of primes, you would run your computer for a provably finite amount of time and establish finiteness for this finite set, too.

No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically by performing $p^n$-descent for higher and higher $n$, until the upper bound on the rank of $Ш(E/\mathbb{Q})[p^n]$ stabilises. Of course, we cannot prove a priori that this would ever happen, but in practice, if you knew finiteness of $p$-primary parts of sha outside a finite set of primes, you would run your computer to do $p^n$-descent for the remaining primes, until you establish finiteness for this finite set, too.