Dare I even say this? Much of Paris is not necessarily touristy, but as travelers, we tend to gravitate to the famous sights, the comfort zone areas of this grand city that enable us to say, "Oh, yes, we went to the Louvre, but it was so packed with tourists! But we did finally see the Mona Lisa--you know it is really small!"
Some of the "outer" arrondissements are the richest insofar as the cultural experience is concerned. I, too, love the 10th. And the 11th. True, there are fewer world class tourist attractions, but I will never forget a morning spent in a public park in the 10th, watching French school children playing on the climbing apparatus under the supervision of their teachers. As I nodded off due to a persistent bit of jet lag, I thought, "this is what France is really like."
A stroll in the Luxembourg Gardens, or along the top of the Promenade Plantee, or even along the city streets connecting the Ecole Militaire and the Place de la Bastille (done without a clear goal in mind) are head clearing activities, and allow for a connection to be made between you and the City of Lights. I agree that off beat places like the Pere Lachaise Cemetery and the banks of the Canal Sainte Martin offer an immersion into the reality of Paris. Just about anywhere you can stroll to will allow you to capture a Metro to take you back to your place of origin.
See Notre Dame, by all means, but flee the crowds and find yourself on the streets as a Parisian rather than a tourist captured by the recommendations in your tour book.
Edited to add: By all means, with your limited time, skip Versailles. Much more relaxing and satisfying things to see downtown.