I've never worked in a critical access hospital; the closest was my first CNA job which was a single building with 1 hospital floor, 2 LTC floors, and an OR. I believe it was considered a CAH.
But no, I have never heard of having to find your own replacement. What are you supposed to do if nobody wants to work -- come to work with what could potentially be rib fractures or a pneumothorax, and/or a TBI?? Or norovirus or the flu or......??
Even small hospitals have administration, and it should be THEIR job to staff appropriately. The fact that we too can get sick or hurt is not new information, so why don't they have a plan in place? At the nursing home in my first paragraph, if nobody was able to work, the ADON put on her whites and worked (there nurses wore all white) -- or colored uniform if she was filling a CNA hole. She was very clear that as a nursing leader, this was her job. Or they could make an on-call schedule where a nurse is paid to be on call, and called in if needed (of course this would require more staff, so that the on call RN doesn't go into overtime or work >12 hrs per day.)
But no, it should NOT be the staff nurse's job to be schedulers on their own time. Those hours in between the sick call and the start of your shift is your own, i.e. unpaid, time. So why in the world are you expected to perform a scheduler's job function for free, while off duty -- especially when a sick nurse should be seeking treatment or resting!!
And.....I just read your post again and noticed that YOU HAD A ******* LOC!!!??!?! I read quickly and missed that. Please focus on getting better!! My maternal Grandpa and Great-Grandpa were both dairy farmers (of THEM, my uncle survived a farming accident, and a great-uncle died in a farming accident), two of my best friends were dairy farm kids, my parents' neighbors are dairy farmers, and I participated in the equestrian sport as a kid/young teen. When I was 12 I got kicked in the ribs by a horse I had spooked.
Oh, and I went to school with a girl whose father survived being gored by a bull. Those big muscle-bound animals are NO JOKE!! They can KILL PEOPLE!!
I am so livid for you right now.