While I'm sure ther are multiple places you could do both Semitics
and Classics, I'll mention two. First, the Univ. of Calif. at Berkeley,
offers PhDs in both Classics and Near Eastern Studies. GTU, which is
next door, offers a joint degree program with UCB in Near Eastern
Studies, in which you also take some OT courses.
The Univ. of Calif. at Los Angeles also has a very good Ancient Near
East program and has a Classics program as well. You couldn't go wrong
studying ANE languages at UCLA. I know there's a serious Classics
program also at UC Irvine, but I don't think they have a Semitics program
(UC Irvine is where the TLG comes from).