I think you have two levels of crowds in Rome -- very crowded and
very, very crowded. We have been in Rome in May, June, and October
over the years. About the only difference we have noticed is that it
is more crowded in June BUT it didn't seem to that much more crowded -
just very crowded. We enjoyed Oct the best but that was mostly for the
weather but still crowded. Rome is the mecca for the Christian world
and it just attracts a lot of tourists and most are non-Americans. I
am not sure that there is an off season to Rome. But -- I am not sure
what Rome would be if the crowds were not there. Good luck.
To respond to Frank's above quote--yes, we experienced what one would call 'off-season' in Rome, and that was February 26 - March 4 of this year, in beautifully temperate weather, 50's & 60's during the day, 40's at night, only once a sprinkle of rain.
We stayed at an apartment above Campo de' Fiori, and while the market was busy every day, it never got too bad at all, and by the late afternoon the vendors had left and it was gloriously quiet. The buses we took every day were routinely crowded/full, but I expected that. Made restaurant reservations, some on the day of, some in advance, with no problem. Many nights we walked right into a place and were one of maybe two or three occupied tables for our entire meal.
Attractions were the same. We got advance 'skip the line' tix to St. Peter's but had we not done so, I estimate we would have only waited about an hour. Through Roman friends, found some under the radar places like the Baths of Diocletian--right next to Termini--that was pretty much deserted, and we had the run of the place. Villa Torlonia (the wartime residence of Mussolini) was similar. Walked by ourselves around the perimeter in the quiet dusk and imagined Il Duce standing on the portico. I will say that once we exited the Metro right near Spanish Steps and promptly got the heck out of there, away from the throngs. A few minutes later we skirted Trevi Fountain & exited there in short order as well.
That's the thing about Rome that's unlike Florence--if you see a crowd you can always get around it and go somewhere else that's decidedly quieter, at least the time of year we were there. Kathy & Chani can attest to this.
I see Sandra's downright scary post above and the only answer I can give is that finally the cheap dollar to Euro ratio has caught up to American tourism. I can truly say that I lived a wonderfully chill existence, thank you, for a week in Rome, and it was done cheaper than I could have in my own suburb north of Chicago, and absolutely way cheaper than the city of Chicago itself.
Enjoy your planning!