The behavior you are seeing is, unfortunately, the nature of Windows audio.
Windows audio by default "mixes" audio from different sources and sends it to a single output device. In this case, JRiver MC and YouTube (for example) could both have access to the Mojo. Unfortunately, when Windows does that, it is not bit perfect. That is, it changes some of the bits in the audio. In most cases you often do not hear much difference. That is what you get with the Direct Sound option.
However, with DSD, you have to send the exact bits to the DAC, without any changes, which includes even any volume changes. Any changes destroys the bit pattern of DSD and makes it not work. Therefore, for playing DSD, the driver needs to have "exclusive access" to the Mojo, which means only MC can access the Mojo when MC is running. Unfortunately, DSD requires bit perfect audio and therefore requires exclusive access.
There is an option in MC to funnel other Windows audio through MC. That option is called WDM. You have to make sure it is enabled and you then set it to the default audio device. Then any audio will go through MC. That allows you to send audio from other software to the Mojo when MC is running. One advantage is that you can then use the various DSP options in MC for those other programs. There are some limitations, but it works for most programs. It does mean you have to have MC running at all times. And, you have to be sure in MC that the output audio device is set specifically to the Mojo and not to the Default audio device. There is a discussion of using it in the Wiki.
So, you have two options : either shut down MC to use other programs or use the JRiver WDM option. Personally, I just shut down MC if I am using other programs, but it depends on your usage pattern.