At 65 you can get both the senior freeze and the senior exemption. Typically saves about 1000 give or take off most taxes per year that I have seen. And yes - newer homes for the most part are higher taxes because they cost more and are worth more in the market. Usually. A great deal of taxes are the area though and the schools. If you want lower taxes than Tinley and Orland, you need to go north into the Oak Lawn area.
I would be happy to help you find what you need, considering the taxes. If you let me know what you want, I can search on that basis.