Yes, you can

You can change your J1 to H-1B with DOS recommendation letter. My university also said that they need my J-1 waiver final approval to file my H-1B petition, but, I told them that so many people got their H-1B approval just with DOS letter, then, they filed my H-1B application just with DOS letter, and I got my H-1B approval last week (just took 2 days after INS received my application document) with premium processing). So, just try to talk to your university's international office. good luck!