Take a breath. Nothing has to happen in the next ten minutes.
If your pet died at the vet's office, the clinic will keep the body and walk you through the options when you can form sentences again. You can go home. You can call tomorrow. That part is handled.
If your pet died at home, you're the one handling it, so here is exactly what to do. Their body may release fluid after death. That's normal and it isn't a sign anything was done wrong. Lay them on a towel with a garbage bag or waterproof layer underneath. If you want them resting in a curled position, tuck them that way in the first couple of hours, because the body stiffens after that and stays how it lies.
Then move them somewhere cool. A basement floor, a garage, the coolest room you have. In summer this matters more and the clock runs faster. A body kept cool buys you about a day to make calls. It does not buy you a week.
And before any of that, if you need to sit on the kitchen floor with them for an hour, sit on the kitchen floor. The timeline is real but it isn't instant. You get the hour.