The closest, newer neighborhoods to your husbands work don't have a good elementary school. To get into Athens-Chilesburg Elementarty (ACE) you would need to be in Andover Hills, Brighton Woods, or a very tiny section of the oldest part of Chilesburg. All of those houses are more than 10 years old and typically have the decor you don't want.
The good thing is that your husband shouldn't have too much of a commute from his office to about anywhere in Lexington except for the total opposite end of town.
I deal with a lot of buyers, and it seems most either end up around Hamburg (which would be closer to your husband's work) or the "South Side" of town, which is Nicholasville Rd/Harrodsburg Rd. For the south side, I think Willow Bend might be a good match for you. The houses are newer, in your budget, and just about any neighborhood on that part of town has a good school district. Another good pick would be Dogwood Trace/Dogwood Glen.
For $250k in a newer house, you should be able to get into about a 2300-2500 square foot 3-4 bedroom house without a basement, or less above grade square footage with a basement.
It's been my experience that any neighorhood with a good elementary school district has lots of kids. The elementary schools most of my buyers seem to favor are Stonewall, Clays Mill, Veteran's Park, Rosa Parks and ACE. I'll also mention that Cassidy is a good one, but that district serves houses that are much older than you want.
When you look for houses, use
The Lexington Bluegrass Multiple Listing Service. That is our local MLS. All the other real estate sites are fed our listings, so you may as well go the the source. I have found that sites like Zillow or Trulia don't always take the sold/pending listings off. I've had people say they found a house on Zillow and I look it up only to find that it closed months ago. Also, Realtors are notorious at getting the school districts wrong. It's like they must just assume that the closest one is the right one. Use this link to check that the info is correct:
FCPS Street Directory - Find your school by street address
While we are at it, don't forget to check the crime map and the sex offender registry:
Citizen's Crime Watch Home Page
Kentucky Sex Offender Registry
We have parks scattered all over town, cub scout meetings all over town, lots of private/city/church based sports programs.......I think you'll like it here :-)