I understand now that there is nothing wrong with my DHO804, now I am trying to locate the source of the noise. It feels inconsistent. I tried shutting everything off in this room, which meant unplugging everything except my scope. No significant difference on the original nor a nearby wall outlet. An outlet in the next room seems better but an extension cord back to the original location seemed maybe a little better but hard to tell. I clearly need to find a way to measure this better so that I can more immediately see if it's improving. It doesn't help that I am still learning how to use this scope and haven't had much time to spend on it.
I tried FFT again, and this time I found something different. I set the trigger point to just a little left of the screen and changed the horizontal scale so that I was just looking at the top part of the pulses. Note: I definitely don't know how to use FFT properly, however this time I found a collection of peaks which, unless I did something very wrong, seems to be the sources of my noise, at least today. Also I've moved the scope about 2 meters from where I had it on my desk due to something else I was working on and in this orientation/position the noise is less than half of what it was before, only ~150 mV. Here were the results.
If I haven't made any glaring errors, my problem seems to be coming from a few locations with line of site through my window from 5 to 8 KM away. All (most) of the FM and TV transmitters in my city are arranged along a long hill across the river valley from me, all visible from my window. Could this really be the source of my noise or am I have I really messed up how I am using FFT?