It's a very tough situation you're in. I think it's a shame that you are using YOUR sick time and YOUR vacation time for child care. You're not getting the vacation you need - real down time with no responsibilities. And I understand about the holidays - I always used up 2 in the fall and 1 at Passover, although for many years I worked at a number of Jewish agencies so there was no problem. (But we "lost" other holidays to make up for it.)
But I also see the issue for the employer. One employee (you) is out more than anyone else, and that can create resentment among other employees. They are somehow making arrangements for child care, or they are penalized for not having kids to begin with. You're paid to be there. They are calling you in to HR and writing you up, not because they don't believe you that your kid is sick, but because they don't think it's their job to adjust for you constantly. It is unprofessional. It's good that you can make up the work quickly, but that's not an argument you really want to make publicly with them, because it pretty much says the work can be done in fewer hours than you normally put in. So there's plenty of reason to downsize you to fewer hours all year long, which cuts into your pay every week that no one is sick.
You have 2 options - and maybe you should do both. You need a nanny service of some sort - consider colleges with an education or nursing program, or one with a CNA or PCA program. When my mother needed personal care, I went through a service (and paid higher rates of course) but many of the personnel were studying to be CNAs or other professions, and they were pretty much paid companions (non-medical). They helped with any meds she needed, helped cook/prep meals, took her shopping, escorted her to activities, etc. I have a friend with a handicapped daughter and she hires helpers for a period of 5 hours, but at least 2 of those hours they don't have anything to do and can use the time for studying or reading. Sometimes you can get away with a lower hourly rate if some of the time is "down time" and not active care. Or you need a neighbor, or perhaps a senior citizen who wants an occasional "gig" and who is available on short notice. I also know some people on disability who can't make a big salary or much of anything, but who can work for cash. The problem with cash is, you can't deduct it as a child care expense if they aren't declaring it as income.
The other thing is immune system support. I have worked in this field for years and do a lot of education around it. It's food based and isn't usually something you work out with a pediatrician (they tend to deal more in pharmaceuticals), but I know none of us has been sick in 8 years, including my son. Don't go for the junk on the store shelves though - some of those companies have been caught (and fined) by the FDA for bogus claims.
If you go look for another job, how will things be different? They're going to have a sick leave policy and you'll lose any seniority you have accrued now. And a new company will be very unlikely to be so understanding if you are calling in sick right away. You and your husband need to work a deal with your employers or find jobs that involve some work-at-home time if you aren't going to get a short-notice sitter or do systematic and comprehensive immune system support. I wish it were different, and I do sympathize with your dilemma, but it's the reality of the world.