Yes, most people DO apply right after finishing their junior year, or at least during the summer before beginning their senior years.
If you list on your AADSAS application that you're working towards a BS or BA degree, then 95% of the schools will expect you to complete the degree BEFORE matriculating. Meaning, you can't just get accepted then drop out of school.
The more important point is that schools want you to be true to what you list on your application. If you don't list that you plan to graduate (that is, if you know and the schools know that you plan on matriculating without a degree) then schools typically don't care about you finishing up classes because they weren't planning on you entering with a degree.
Does that make sense?