Your experience is the undeniable trumpet of truth. Without a doubt there are homes existent at your price point that rest essentially in the quadrant with boundaries of Irving on the south, Foster on the north, Ashland on the east and a montage of Western slurring over to the river on the west. I'm pretty sure your experience parallels that of my recent and past clients looking roughly in the same area: slightly or negligibly rehabbed properties with "knock your head" clearance in the basement and garreted attic redos with slant down ceilings to inhibit furniture placement on two walls.
There is nothing wrong with this inventory, but if you seek something else, like a full on 4 bed/three and a half bath model with finished basement where you won't wound your noggin when you descend the stairs, you may need to alter the trajectory of your search.
As at least one earlier respondent commented, find a credible real estate professional. If it's me, great. The choice you make is pretty important though because the agent will be able to from top of mind awareness discuss with you all inventory that is currently available, mention the pluses and minuses, let you know if you are in Waters, Coonley, Bell, McPherson or Ravenswood. Share with you additional inventory that you might not have considered (such as considering options in Edgewater or dipping into Albany Park or veering northwest into Sauganash). And always offer opinions rooted in you making the best decision for the best house at the most commodious price.
The same agent also needs to be conversant to let you know if your price point and aspirations are mercilessly colliding with the "NO" of a location. When I run a cursory search with your parameters I detect 8 homes that more or less meet your criteria. However, every one of these homes represents to a degree my description above, particularly with the upper level ceiling necessitating furniture placement on either this wall or that wall but not on either side wall.
So where does this leave you? If you are adamant about Lincoln Square, some degree of compromise is critical. If the house matters more than the area I urge you to select the most knowledgeable agent to serve as your resource to 1) identify prospects, 2) personally view homes with a critical eye to finish, size, price and location, 3) negotiate on your behalf fully edified as to market analytics, 4) reach agreement (or walk if agreement is undoable), 5) provide you with strong options should you need them for inspection, attorney, lender and trades people if necessary, and 6) see you through the process in a seamless and professional manner always holding your interests as paramount.
And if the home is more important then the agent needs to do all of the above while charting your itinerary into new locales.
I wish for you the best of fortune as you seek your home.
The Real Estate Lounge Chicago with @properties
Expertise in Negotiation, Buyers' Representation and Luxury Home Marketing