That is so odd, I find cauliflower lasts for ages in the fridge. It might get some funky spots on the surface but I shave those off with a veggie peeler and the rest is fine. I keep it wrapped in the cello it comes in though or it will go limp very quickly.
I think sprouting is the way to have fresh greens all the time. It is very easy and very small batches can be done every couple of days so there is always some just perfect for eating and not much or any left over. Alfalfa or clover are the easiest but lentil sprouts are super easy and Very tasty. Sunflower and buckwheat "lettuce" or baby greens are more involved but also doable with a couple of small shelves and either a very sunny window or a small full spectrum light. The alfalfa though is the most simple and satisfying as it will green up in even dappled natural light in a day or two.