I ended up on the embedded side from an EE background (which is actually a fairly common route).
What does "Full stack" mean to you? Right the way from configuring the MAC hardware and the handling the MAC buffer received ISR up?
Have you done much with writing device drivers? A lot of embedded development is that sort of style of code.
Embedded these days is a somewhat broader church then it used to be, but usually it still means working close to the metal, usually in C or similar (Sometimes these days a very restricted subset of C++ is used) in more or less memory constrained environments (Code space of a few kB to maybe a few MB or so, RAM from maybe 64 bytes to a few MB or so).
Assembler is something we use reluctantly, but it does happen on occasion (Usually in the context of interrupt jump trampolines and such).
The embedded crowd tend to expect familiarity with the use of oscilloscopes and logic analysers to debug things, and knowing your way around such things as well as how SPI/I2C/CAN/UART/Modbus interfaces usually work on the wire is more or less expected in many jobs.
Sometimes there are very restricted subsets of languages used, MISRA C for example being a 'safer C' popular in automotive use.
At the top end you find the sort of multicore ARM that are used in modern smartphones with all the trimmings (Including Linux!), but there are probably ten 16k ROM, 2k RAM projects for every big system (And the big systems may as well be PCs for all practical purposes).
By all means go for it, it is fun shit to work on, but do expect to be getting down and dirty with the register level stuff and to be reading datasheets and silicon errata on a regular basis.