I think Edico is almost correct. I believe there should be a "may" before decay, because there's a chance of decay, not a certainty. You can, for instance, have a stack of roasted meat cuts and a partially eaten one, and see only one decay.
My first priority is tanning, so once while the skin is going through the first and second tanning stages I take care of the meat (and have it stored in a cellar as much as possible). This frequently means I'll start drying of a few batches after starting the first tanning stage, go to sleep, start the second tanning stage, and then take care of the rest. Tanning takes a long time, so there's a fair bit of time to take care of the meat.
I don't hunt far from home. If I see game while on a trip I'll ignore it, as I don't want to be tied down taking care of he food/skin for days. If on my way back I may very occasionally go for some game where I can carry everything back home (which shouldn't be further than the rest of the day away in summer).
If I were to hunt near a village I'd use their sauna to smoke the meat (or build a shelter to dry it, if the season is suitable). I've heard villagers make take some of your smoking meat once it's prepared, but it shouldn't be a substantial loss. My preference is always to dry/smoke meat, saving only as much fresh meat I need for cooking (using the Njerpezit Cooking Mod). Spoiled meat can be used as dog food, as long as it isn't prepared.