I feel your pain, source of anxiety for me too.
I can share what I've done and plan to do. I haven't sent out postcards yet but have sent email blast to publishers with 8 to 10 images, funnily enough I got out of possibly 100 emails sent out, I recieved around 15 responses (rejections) some positive comments. A year and a half after sending an email to and Irish publisher, I received an email, out of the blue, asking me to submit samples artwork for a project, they liked it and I Just finished up the Job, first Children's book for me and through a cold email.
I'm Irish myself (working out of Japan) so I was targeting Ireland and the UK. I purchased the 'Childrens's writers and Artist's Yearbook' which has a directory of agents and publishers and a few lines of their general style they do which was helpful. I also joined and purchased directories from the Association of Illustrators in the UK. I made my way through the lists researching each one painstakingly making notes of where I thought my type of work could fit and compiled a list with dream clients at the top and the worked from the bottom up. I plan on sending out postcards soon.
There is a video in the SVS class regarding mailers and postcards and about sourcing addresses etc. title is how to get freelance work. I remember watching and thinking there was really good info in there, like do you send multiple postcards to one house etc. I sent multiple emails to one publishing house, not sure if that was a good idea. I got a lovely response from Random House Art Director saying she would show her collegues my work and was reading books and I may fit with one of them, never heard and thing back after that but that response had me floating haha.
Best of luck