You would not convert your L-2 visa. You would simply apply for an F-1 visa. Either option you have presented has risks. Your university has already provided guidance about the negative in applying for a student visa and that is administrative processing. You could find yourself waiting for months to have a student visa issued if your case is stuck in administrative processing. This can delay your studies. There are no real benefits to obtaining a visa now other than you would have the visa in your passport for future international travel. Administrative processing occurs when a visa decision cannot be made. There are a variety of reasons a visa application ends up in administrative processing from request for additional evidence to delays in the completion of security checks. An additional issue that may come up is immigrant intent if your spouse or parent is in the process of obtaining an immigrant visa through employment. An F-1 visa unlike an L-2 visa does not allow for dual intent.
Answered on May 23rd, 2016 at 2:31 PM