I think a job where it's understood that you may have to take a day off with very little notice, or more, will go a long way to everyone's peace of mind, and I found it was possible to find that.
A board and care home can provide full time basic care, usually with monthly "rent" for the full package. Respite care is a shorter term variant for when a caregiver needs a break. These are not nursing homes because they lack staff doctors and registered nurses. They have a little economy of scale by sharing caregiving staff with several residents.
There are also hourly/daily caregivers who specialize in providing in-home care. This can get very expensive with frequent use though. There are less specialized variants for "sitters" and transportation escorts who can be a companion but don't handle as much of the patient needs.
Try to check with your hospital system or county social services, as they can probably help you bootstrap your knowledge of these topics. There also may be local support groups where other caregivers may have relevant experience and tips to share, as well as camaraderie.
Prior to the need you might be already planning the next job move. Now you might be planning the next decade at the same employer because of how they treated you.