OP what is your budget and what size house do you need?
I live in the area and know Dulwich Village, East Dulwich, Herne Hill, Peckham Rye etc very well.
Dulwich Vilage = Very established, lovely, green, conservation area, great family homes and populated by families, but expensive. Near to all the schools you mention, very near Dulwich Park, Dulwich Picture Gallery. A small but nice strip of shops that include the local pub ("The Dog"), a few family friendly restaurants, convenience stores, hairdressers etc. Only a 10 minute walk to Lordship Lane so benefits from East Dulwich's coolness. 10-15 min walk from Herne Hill station. You'll struggle to find a family home for less than 1mm.
East Dulwich = much more urban and trendy, fantastic high street on Lordship Lane, loads of independent shops/bars/cafes. Extremely family friendly ("nappy valley"), however prices have rocketed recently and you won't find a 3/4 bed family home for under 600k. The houses tend to have quite small gardens (yards).
Herne Hill = Property cheaper than Dulwich Villlage but has a lot going for it. Adjacent to Brockwell Park. Great local shops/cafes/restaurants and a great Sunday market. Excellent transport links (rail) to central London. A bit more low-key than East Dulwich but has its own community feel which is more established (in my view) and quite nice. Probably less of a transient population, many families move out of ED when they outgrow their space and find they can't afford the prices for a bigger home. Herne Hill is very close to Jags/Alleyns.
West Dulwich - I don't know this area as well, but it's worth checking out as it's cheaper than Dulwich Village but is still fairly near your schools. Personally I prefer ED/Dul Vil or Herne Hill.
Peckham Rye - Quite urban like ED but great (better in my opinion) houses and gardens around the Bellenden Road area and extremely up and coming. Minutes from Lordship Lane, excellent transport. Not too far at all (~20 min walk down East Dulwich Grove) from Jags/Alleyns. Great bars/cafes etc, notably the Victoria, Montpelier plus numerous other places opening up on Belenden Road. A bit of a hike from Ducks though.
Sydenham Hill - much more suburban but if you work/teach locally it would be a very easy commute. Plus probably cheaper than all areas above.
You should consider what your back-up options are if you don't get into Jags/Alleyns as state schools are very over-subscribed and these two schools are highly sought after and hence selective with no guarantees of getting in (unless as a member of the teaching staff you get an advantage?).
Good luck with your move, it's a great area.