Both of those towns would have what you are looking for. Here are some listings that fit your criteria:
RealtyUSA - Property - Property-List
These listings are in the Northern suburbs and within the North Syracuse, Liverpool and Baldwinsville school districts. You could find something in other districts within 20 minutes of his employment as well. I'll try to add more later or hopefully others will chime in.
Just curious, but would your child be old enough for a Pre-K program? North Syracuse has one at its Main Street School in the village of North Syracuse.
Another district that came to mind in terms of newer homes and lower taxes was Central Square. It would be within 20 minutes or so of his place of employment and it is an area that has had parts of it grow. Here are some listings that meet your criteria:
RealtyUSA - Property - Property-List
In that district, I would look into listings in Cicero, Clay or Hastings, which is just inside of Oswego County. Brewerton is in that district and is in both Cicero and Hastings. It is close to I-81 and a relatively quick drive down Route 11 from shopping in Cicero.
Here are other listings that fit your criteria within Onondaga County:
RealtyUSA - Property - Property-List
I searched for homes from around 2000 and on.