I love the idea of scouting and recruiting both being available during the season but I don't like the rigidness of limiting it only during the season. Recruiting/scouting should be fluid and available both in season and off season.
Also, in order to address the issue of moving to a new school and not getting any of your players until the second year, maybe limit the number of in season (or early period) signings to a max of 2 per team, then they have to fill the rest of their openings in the late signing period after job changes, transfers, and early entries occur? Also increase the number of transfers from a school if the coach that recruited them leaves. That would give the new incoming coach a chance to recruit their own guys during the late signing period. Now the complaint here would be that a lot of the good recruits would be taken by that point, but 1) that would be exactly like real life where only 20-30% of the elite recruits sign in the late period, and 2) could be offset by the late signing period being when ineligibles could switch to being eligible. And there should be a change in the frequency of ineligibles becoming eligible. It doesn't happen enough in the current engine.
I'm also not a fan of the two separate budgets. You should be able to dip into your scouting budget for recruiting purposes and vice versa. Having two budgets is just creating something unique in the game not reflective of real life. If I'm a coach with one open scholarship and I have a $10k budget and I allocate $5k to scouting, $5k to recruiting, but I find the guy I really want after my first camp, I'm going to throw the remaining $9k or so I have at him not limit myself to holding onto $4k of scouting money for no reason at all.
I also like what chapelhillne suggests above. The Favorite school mechanic needs to be drastically overhauled. New players could be generated with a list of favorite schools through some sort of algorithm taking into account the recruits location, skill of the recruit, school prestige, recent success, etc.
Also, I'd suggest keeping FSS around but only as a scouting tool for the late signing period to allow new coaches who didn't hold camps etc. to get a feel for undecided players that are available.