To be clear though, hormones don't define how your body functions, that'd be DNA.
Chicken or egg. Hormones don't work without DNA, but also your cells don't function without hormones. The DNA has the instructions, but the hormones tell it what instruction to follow. Your cells don't know to create a leg where your leg is until hormones tell them that they're supposed to be leg cells. The DNA is the same, but the hormones tell them what to do.
You wouldn't have hormones or the ability to process them into meaningful signals without DNA.
Sort of true, but we can now add/change hormones that aren't created in our bodies by design. We also get hormones through our diet too. DNA does have instructions that allow it to produce hormones when the right conditions are met, but that's not the only way to get them, nor does that define sex as we all produce both sex hormones in different quantities.
but it does have an objective, scientific, apolitical, societally functional meaning.
That's the funny thing; does it? We all think it does, until you try to define it. Give it a shot. If it's XX/XY, intersex people break it. If it's reproduction, sterile people break it. If it's what gonads you posses, people who have had hysterectomies break it.
I've never actually heard a definition that doesn't break down somewhere. It can still be useful, but it's a spectrum. If you define it as a spectrum then it's actually pretty simple, and it's based on how their body is currently functioning.