I've been writing C on and off for 27 years or so and I still enjoyed this book from the list on your stackoverflow link:
21st Century C - Ben Klemens
http://www.amazon.com/dp/1449327141/?tag=stackoverfl08-20
You asked about "Modern C" and that caught my eye. Most instructional resources on C focus on facts about C without putting them into perspective as to which C era they belong to. "21st Century C" takes an explicitly modern perspective, is opinionated as to which aspects of C you can postpone or ignore, and provides updates for people (like me) who have mostly been familiar with K&R or C89. I found it a really fun read and would definitely recommend it as #1 for your question about "Modern C".
As to if you should learn C, I am super opinionated but I think C still has a lot of value, even for someone who doesn't describe themselves as "full-stack". The reasons are obvious: the kernel is written in C, libc is written in C, the webserver serving your web application is almost definitely written in C, but if it isn't, the compiler that compiles the language it is written in is almost definitely written in C (hats off to Golang for getting off their C compiler). Despite its flaws, I love C.