I believe there is an increase in children with anxiety problems, mostly because of the way in which they lives their lives today. Everything, but everything is so much more goal-oriented, and there is much more pressure put on children these days in terms of education, endless tests, exams etc. I also believe social media has played a large part, and that the less active lives people live mean that anxiety can build up instead of being released in harmless ways.
Have you asked the 9-year-old what is making her anxious? Does she enjoy her lessons? Can you work out why she feels a need to put pressure on herself? Gently teasing out some of the answers (perhaps over a period of time) may encourage her to open up. Does she feel this way at school / home, or just in music lessons? You could try helping her to see your teaching room as a "safe space" and encourage her to enjoy what she plays for its own sake. Maybe introduce a fun element or two (I'm not suggesting at all that your lessons aren't fun!) just to help her relax and laugh. Does she have any underlying health problems (anxiety disorder, Asperger's, etc, etc). The finger-chewing may be an ingrained habit, or just because she needs to do something with her hands in the gaps when she is not actively playing. Could you suggest something else for her to fiddle with that is less damaging / painful - eg a stress ball or other item?
The 13-year-old does, as you say, have a bigger issue, and from what you say it sounds as though the parents are aware of it - and hopefully taking appropriate steps. It could be that off-loading to you is one of his forms of release and may be very necessary in just helping him to negotiate life. As long as the parents understand that it eats into lesson time and is the reason that he is not really progressing, it seems you are doing all you can.
I live myself with an anxiety disorder, and understand how debilitating and overwhelming it can be. Finding "escape hatches" is just one important way that can help, channelling their energy into something that they can get absorbed in. It might be that these two find you such an "escape hatch" and as long as you feel able to work with the limitations this imposes, I don't think you can do any more for them without stepping outwith the boundaries of being a music teacher.