The revelant information is in the article of Hesselink: Polarizations in the classical groups. By the results of Fu, for any Richardson orbit, you need a polarization (a choice of $P$ for which the orbit is open in the image of $T^*G/P$) such that the statistic Hesselink calls $N_1$ is $1$. By Corollary 7.6 in Hesselink, this number is independent of the choice of polarization and only depends on the orbit. There's a bit of a blizzard of combinatorics, but if you look in the tables at the end, you'll see there are examples with $N_1>1$, and thus no resolution in all non-A types he works out.
It looks to me as though any classical group where all orbits have symplectic resolutions must be $SL_n$ for some $n$.