How do I keep a half-finished piece from drying out between sessions?

The dabbing-with-water trick is only buying you minutes because you're rewetting the surface but the water keeps evaporating right back off. What you want is a sealed environment. Get a spray bottle, mist the whole piece lightly, drape it in a damp (not dripping) paper towel or old t-shirt, then wrap the whole thing in plastic so it's basically airtight. Fold the bag over so it isn't sealed tight around the thin tail bits, since trapped condensation there can cause sagging. Check on it every couple of days and re-mist if it's drying out. If you keep doing this you can hold a piece in mid-build for weeks.

That said, for a galloping horse with a thin tail, don't panic about keeping it uniformly soft forever. Something like that is honestly easier to finish in stages. Build the body soft, let the whole thing firm up to leather-hard, then add the tail as a separate piece. Which brings me to your join question: yes, you can attach stiffer clay to softer clay, but it needs prep. Score both surfaces really well, paint on slip (clay mixed into a yogurt-ish paste), press together firmly, and reinforce the seam with a small coil of soft clay rubbed into the joint. What cracks joins isn't stiffness itself, it's two surfaces at different moisture levels shrinking at different rates. The closer they are in wetness, the happier the joint.

Habits from polymer worth unlearning: first, polymer never shrinks, so you probably eyeballed details on the surface. Real clay shrinks as it dries and again in the fire, so thin attachments made late can pop off even if they look fine going in. Second, don't leave thick sections solid. A big solid horse body will dry on the outside long before the inside, and that trapped moisture is what makes pieces crack or worse in the kiln. Ask your instructor about hollowing it before it gets too firm, either by cutting off the top and scooping or paddling the walls thin. And go slow with drying. When you think it's bone dry, give it another week sitting unwrapped on a shelf. Rushing that stage is the number one way first sculptures die.

I'm not sure I agree that the entire piece needs to stay perfectly soft indefinitely before you add anything to it. It feels like you're treating this like a delicate scientific experiment instead of a sculpture. If the main body of the horse is sturdy and you manage the internal moisture so it doesn't crack, a little surface hardening shouldn't ruin the piece. It sounds like too much pressure for a first go at this. Maybe focus less on the perfect moisture level everywhere and more on building up the thickness correctly from the start so the internal layers dry evenly. I found that just letting the structure breathe in a sheltered spot worked just fine for my initial experiments.

I've had this happen before, and it's usally because the room conditions are too erratic. You don't just want to cover it; you gotta make sure the temprature around it is pretty stable, too. Don't put it near any heat source or a cold draft, like an open window or an air vent while it's drying. It seems like steady air movement is just as important as keeping the moisture in. When it gets hard, even just a little, it'll get brittle, so you gotta be extra careful handling it.

I'm actually working on a small bowl instead of a figure, so I'm much more worried about the bottom drying out too fast. The talk about how crucial moisture is realy clicked for me. When people mention it reaching 'leather hard,' I just picture it being stiff, but I'm unsure if the clay feels different when it's properly ready to be handled without major issues. Does it feel smooth, or is there still a slight grit to it when it's supposed to be at that stage? I don't want to mess up the curve before I can get it glazed or whatever.

Update: the mist and damp cloth wrap worked. Covered the horse that way overnight and it was still workable in the morning, finished the tail with no drama. Also moved it off the shelf near the window after reading about drafts, which probaly helped too. It's drying evenly now, fingers crossed for the kiln.