I'm not sure I'm the best person to answer, but as no-one else has - I will start you off
I did my Degree in Graphic Design & Illustration, though I went on to work in fashion as that was the trade I grew up with - rather than advertising or illustration as such, but my work did often marry the 2 with fashion illustration & I have done quite a bit of other illustration work over the years, but I have been asked to do it based on work they have seen of mine, rather than any qualifications, applying for jobs etc, things such as illustrations for fabric design houses, or screen print companies for t-shirt/top or placement print designs, so my graphics back ground did help there, but I was never asked for qualifications - they just had seen work I'd done & asked
I've friends who have done medical illustration for books & again it was down more to being in the right place at the right time & being asked, rather than the Graphics degree they had, but I think that helped - another friend has illustrated childrens books & her back ground is in animation -
I suppose from that, having a sound knowledge of the printing process involved helps you to know the restraints that you have to work with, which any potential employer will be able to spot in your work.
0ther than that just keep sending off examples or as a Animator friend did - work for for free for a while to get a foot in the door & prove your worth, so that when paid work comes up, you get it - he's now an award winning animator, with a fantastic work portfolio & gets paid big bucks for doing something he loves
I did a two year foundation course then an Illustration degree. But i've never been asked to prove i did them iyswim - most clients go on portfolio and word of mouth. If i'm honest i didn't learn anything much form my degree (foundation was a very creative time that i really enjoyed) and came out none the wiser about how to set up a portfolio or how to approach possible clients.
I now have a website which i routinely send off to anyone i think might be interested. I have sold work to go on peoples walls too - in fact that's the way it seems to be going for me at the moment...