You are confused. There is no common app. All apps are individual school apps.You have to sign up with LSDAS so that each school can receive your transcript and LSAT info (you can also send them your LORs if you want).That said, LSDAS also offers electronic versions of each school's individual app, and you can fill those out and submit (and sometimes pay them) them online. If you apply electronically through LSDAS, there is also a "common information" page where you put things like your name and address so that the infromation autopopulates into your electronic apps. But there is no common app. Also, make sure the info autopopulates right; each app is slightly different, and so you have to make sure everything looks right...Hope this helps.
I only applied through LSDAS, and I don't think there's any downside to doing it that way. Any supplemental or school-specific materials were just attached as word documents. It was super easy this way plus LSDAS keeps everything in one place so if I needed to print out an app later, I knew where it was and didn't have to make copies of paper apps.
When you submit an application through LSDAS it takes awhile (at some places up to 3-4 weeks) for the school to receive your application. Then the school then has to request your letters of rec and LSDAS report, which is another several weeks.
I only applied through LSDAS, and I don't think there's any downside to doing it that way. Any supplemental or school-specific materials were just attached as word documents.
For one thing, most of the time, the system will know which schools have granted you a fee waiver and you won't have to pay.