The intensity changes as I move around the house
Yes, this is normal.
Low frequency sounds create standing waves inside rooms. It's a bit like an organ pipe or a flute, where the acoustic wave bounces back and forth in a pipe. This creates resonances at certain frequencies whose wavelengths correspond to the length of the pipe. The same happens in a room where acoustic waves bounce between walls. Some frequencies will be amplified by this resonance.
Unfortunately, this depends on room dimensions, which you can't change. So if there is a source of noise in the environment that emits a frequency that corresponds to the resonance of your room, you have a problem.
I had this issue in a flat I used to live in, a constant low frequency hum that was very annoying. I traced it to a neighbor's ventilation fan that was clogged with dirt, so it was off balance and it vibrated. Cleaning it fixed the issue. Due to acoustic resonances the intensity of the hum would vary wildly depending on where I was in the building, so it was impossible to find it by ear. I had to stick a stethoscope on the walls to listen to the noise, and that helped locate the source.
You could record the hum with a good microphone to check if the frequency corresponds to mains, or a multiple of mains. But if you still have hum with the power cut off, then that may not be fruitful. If it is constant, then it is not the wind.
You can try opening and closing doors inside the house to see if that changes the hum sound. This changes the volume where standing waves can resonate. Also, the air inside a corridor can act like a mass/spring and resonate. So the solution may be just as simple as closing a door, or putting a curtain across the corridor that resonates.
If that doesn't work, and if you can't shut it down at the source, then you'll have to setup some bass traps.
First, try to measure the frequency of the hum. You can do that byear, or get/borrow a good microphone, record it with a computer, and use audacity's spectrum feature to get the frequency.