What you're doing doesn't sound sustainable at all. The long commutes and work times, along with the expectation that you then cook meals. Something has to give. Why can your partner not do more in the way of cooking etc? Is there a chance that you/he/both of you can do some simple meal prep/freezer dump bags so that supper is cooked quick (or bunged into a slow cooker and ready for the evening)?
The expectation to stay beyond your shift time is nuts. Do you get paid overtime for this? What would happen if you worked to rule/finished at your actual finish time? Is there a way to do some hybrid work/WFH? Or compressed hours over fewer days?
FWIW, I'm a nurse, so I understand completely the shit show of juggling work, the long hours etc. I also totally get the mum guilt: when the DC are young, you do feel like you're running about like a blue assed fly. I'm a single parent with two DC and it was bloody hard going a few years back (they're teenagers now, so different set of concerns and worries). I also understand the battles with a child that is super fussy - if it makes you feel any better, my youngest was the same (bloody nightmare!). But now she'll eat a wide and varied amount of different foods, healthy meals and so on, and gets involved with cooking etc. It just took a few years to get to this point.
Everyone's different, but what worked for me was meal prep done on my days off: I'd cook 3 or 4 different meals (each serving 4-8), divvy them into portions, label and freeze. We'd have them for suppers, lunches (the kids had Thermos food flasks so they could take in a hot meal to school). I'd pre-do stuff like porridge or fritatta, so healthy breakfast were sorted too. Also things like robovacs for doing floors etc, or dishwasher/tumble drier/modcons that made my life easier. When the kids were younger, I was able to work adjusted shift times to allow me to be available for the wraparound care drop offs and pick ups (I was literally on my own with the DC, no family about, exH not fit to look after them unsupervised etc). Once the kids were old enough to walk themselves to school/be left for a bit at home, I was able to re-adjust shift times a bit more. Once they were a bit old, I went back to working usual day shifts. I won't do nights until my youngest is older.
Can you get yourself signed off like @PermanentTemporary and @easterb suggested?