Should I ask for generic reference letter?
Faculty members will not write "brand new" letters for every school and program you want to apply for. However, once the basic letter has been written, addressing it to the intended recipient is not very challenging. So it is OK to ask your advisors to write a letter for you, and to send it to multiple places. (If you're applying to schools in the US, the faculty member will be responsible for uploading it herself in any event.)