I never used just narrow CW mode to look at this signal. Rather, the demodulation mode might be narrow CW, but since pretty much all I use for this kind of monitoring is SDR I have always seen that wider signal you are talking about, I just never mentioned it because I assumed everyone else was seeing / hearing it also. I think you are seeing multiple signals, not just a single one. The wider signal you are showing there goes away for me around sundown, while the ditter stays in there a while after the wider one goes away. 13560 kHz is smack in the middle of an ISM (Industrial, Scientific, Medical) band. These are regions where there are all kinds of mystery signals, pretty much any device that must radiate for non-communications purposes is a candidate for these bands. For example, some (not all) RFID tags and readers work at 13560 kHz. I have seen laser pumps and modulators at that frequency (and similar ISM bands).
Signals seen here might be local, or might be DX, for example I traced down one particular 13560 kHz signal to a local retail store, it is part of their inventory control / shoplift detection system at the door of the store.
T!