I don't think it was a mad rush from DII to D1-AA (I refuse to call it FCS). The rush was from Division II to Division I in basketball, and and a lot of these schools were happy to bring along their football programs because it meant a chance to play some paycheck games. Even withe paycheck games, 1-AA football programs are still losing money, but now they get their name on the ESPN ticker once per year.
1-AA is not just Division II with more scholarships. It's Division 1 athletics for schools that want Division 1 sports but don't want to put 85 scholarships into their football program.
Even within 1-AA, there are basically three subdivisions. Non-scholarship football (Ivies, Pioneer, Patriot), limited scholarship (Northeast, soon to be Patriot) and full 63 scholarship (CAA, SoCon, Missouri Valley, etc.)