Yes, the problem of averages with a large window FFT, is time.
I really like using the spectrograph in my classes to explain the delay that occurs just a filter, and see how energy moves in time.
1. Turn down or off the background noise to lower the noise floor of the room. Can be as simple as turning off the air conditioning, telling the custodian to vacuum the room later, telling the lampies to take a break, etc.