The answer is yes. And no.
Mostly Rome is not very crowded just up to Christmas and on Christmas.
It starts getting pretty busy right after Christmas and leading up to New Years and then through January 6 (Epiphany or befana.)
So for the dates leading up to Christmas you should be able to find inexpensive accommodations in Rome, but from the 26th, expect rates to go up.
After a day of sightseeing in Rome and Trastevere the best place to rest is placed in Trastevere; HOSTEL TRUSTEVER located in Via Francesco Benaglia, 13.