I have the Yeti as the default card, and when I go into JAWS and set it to my PC internal card, I lose speech. Same thing happens if I have the PC as the default, and I set JAWS to use the Yeti: no speech.
When ou lose speech with the Yeti, is it able to play other audio, or does it stay silent? You may want to try changing its sampling rate in its advanced properties and unchecking the box allowing applications to take exclusive control of it.