Now that you need more room ,and schools come into the picture, you may want a new perspective, which depends on a volatile market for construction and, as importantly, mortgage finance. Your asset mix seems to work on a relative price point view.
Let me offer a new perspective, a bird's eye view. Google Earth to 412 East Austin, Libertyville , Illinois and you will zoom in to about 3200' . You will see the interesection of the major roads and can begin to fly around and locate schools and parks. Zzoom in further and you begin to see if you may walk or ride a bike to school, parks and all the other places that are important to you. This particular neighborhood is near in to trains for Chicago and has a nice flow to it. It's quite easy to have an overview of the neighborhood you will choose. Use this way and see the surroundings of places that appeal to you. You will still do a lot of driving around, but if you consider the satelite views you can save alot of effort.
Pricing is in such a state, for many reasons, depends... I use Libertyville as an example as it's around your price point and has such nice appeal for anyone in your position. As for the developers, homebuilders and their agents, let experience be your guide. Speak to a mortgage lender. If you will need to sell your current home, you may need cooperation from the seller. A cash offer shows strength and allows you more room to negotiate your deal. A transitional market has so many variables to consider, careful consideration to price and market conditions, and good use of the terms in the deal will work to your benefit. An old Chinese philospher once observed, there is a great chaos under heaven, and the situation is excellent.
Oak Park/Brookfield/LaGrange areas may also be great. I also agree that Beverly is a prime choice.
Lots of transportation (Green Line CTA in Oak Park and Metra for all). Great Schools. New construction may be hard to find at a reasonable price...this will likely be the most challenging component of your search. New Construction close to the city is rare and therefore, costly. If new is a must, you might want to consider a townhome in the collar suburbs.
Other areas you might consider would be Burbank, Hickory Hills or Bridgeview. There are a few new construction homes in these suburbs for under 550K, If this is your price range.
If you need more information on any of these properties, I can email them to you.