First of all, your experience is a very common one. Internships used to be a reliable way in, but they've begun to dry up. Unpaid internships are an invitation to law suits, obviously, so many studios are banning them across the board. And because California's minimum wage is pretty much exactly what a writers' PA makes (ouch), a paid intern is basically another PA, and assuming there's room in the budget (which there often is not), the spot is likely to go to someone with prior experience. Hello Catch-22!
Here's the bad news: It's possible to go through the entire USC program and never get an internship in a writers' office. (Note my use of the term "office" -- that's a much more realistic goal, especially without previous tv production experience.) It bums me out, because the program costs an unbelievable amount of money, and yet they really don't help students find the kinds of internships that actually lead to post-graduation jobs. I have a friend at Northwestern whose professor actually called in a favor to get her on a show for a summer. That kind of initiative is unheard of at USC, but I admit, if I had to find industry internships for 40-60 aspiring TV writers every year, I'd cut my wrists, so I can understand why the career office and professors aren't chomping at the bit to get people placed. And of course, these openings are NEVER posted publicly -- even putting something on the UTA list generates a thousand resumes, can you imagine what would happen if it went up on EntertainmentJobs.net? Yike.
So it's all word of mouth, which means you have to know someone who knows about these opportunities in order to even hear about them. How can you make that happen? A few options:
The Academy of Television Arts & Sciences places a couple interns in writers' offices every year -- get on the stick and apply to that as soon as it opens. (It'll be soon, if it hasn't already.)
The profs I had, though lovely people, never mentioned internship opportunities to their students, I think in part because if they did it for one student, they had to do it for everyone (see above). So I don't think you can expect much in that direction, BUT EVEN SO: when you take tv writing classes, build a relationship with the professor -- go to their office hours! ask their advice about aspects of your script you're struggling with! come to class and participate in a respectful manner! The goal here is to demonstrate enough maturity and attention to detail that if you ask your prof. if they know any working assistants who might be up for meeting you for coffee, they'll be inclined to say yes, knowing you'll make the most of the introduction. (Also, note my use of the term "assistants" -- junior writers can be helpful too, but assistants are the ones who know about the jobs and, crucially, evaluate resumes from applicants.)
Start reading this year's pilots and prepping for pick up season in May. Get your hands on some Production Weeklys, which will have fax, phone, email for current and in development shows. As soon as new series get picked up, reach out and see if they're open to interns. (Ideally shows that aren't part of an existing empire: For example, a Shondaland series already has an extensive network of assistants and former interns to pull staff from. But something from a first-time showrunner? There's a limit to how many favors that person already owes.)
Give serious thought to going for an agency assistant job after graduation. The big shops do help place their assistants on shows, after said assistant has put in an appropriate amount of time at a desk (18 months is the minimum, I think?) You might think a year and a half is a long time to wait to get the job you "really want" but working at a big agency has prepared a number of my fellow baby writers remarkably well -- they know all the people at all the studios, and all the assistants at those studios, many of whom will be executives right around the time the baby writers will be going out to sell series, etc. FYI, that first job won't be a writers' assistant gig; you'll have to earn that. But you'll likely land as a showrunner's assistant or writers' PA, which is a good start.
Grab every chance you get to mingle/talk/work with the people in the classes above you. Volunteer at the First Pitch rehearsals. These people will go out into the work force a year before you and if you have a good relationship, they can loop you in. (In my case, I knew a grad who became an agency assistant; I asked her for one pilot script and she sent back a zip file with the entire season's worth, network and cable.)
Sign up for a mentor with the writing department when you graduate (duh).