I've run out of stone in Don't starve. What do I do?
Rocks aren't renewable in the open*. When you're playing your first few worlds, especially if you've been unlucky with the random generation, it can happen to run out of resources and die while looking for more. When you'll get a little more experience it takes very little to gather the necessary equipment to build a fire pit and find a little nice place to settle.
Anyway in your current situation you can do some things to cut your losses:
EXPLORE! Even if the islands you've spawned in are very huge, the world doesn't end there. Pack a lot of logs, make a Logsuit and a Spear (these are fundamental to defend yourself from hounds and spiders) and start going around. The swamps may be dangerous, but there's a whole lot of new world beyond them, and you may even find a field of rocks for all your needs.
Research. Especially in your first few gameplays, you should try to research as much as you can. Since your unlocked researches are passed to your next incarnation, you'll have less penalties from your death, and you'll start your new game with some new tricks under your sleeve. Some suggestions: the Fire Pit, the Logsuit, the Spear, the Turbo Farm Plot, the Backpack are all useful and quality of life changing items. (Alas, this part is obsolete; the research points system is no longer in the game.)
Don't be afraid to die. However inconvenient death may be, your new life will restore every resource you've depleted in the current world, letting you start anew.
Don't be afraid to change the world. The first thing you should do is explore your surroundings. You don't like what you see? Reroll. There's no penalty in that, especially if it's your first day, and it can save you a lot of problems later.
Finally, let's take a look in detail at what you can do with the rocks (in brackets the number of stones needed for every item, 1 cut stone = 3 rocks):
Rocks:
- Science Machine (4)
- Turbo farm plot (4)
- Fire pit (12)
- Hammer (3)
Cut stone:
- Crock pot(3)
- Alchemy engine(4)
- Pig House(3)
You said that you've already done some researches, so the Science machine should already be in place. The only other thing that you really need of those above is the fire pit, and you said you have the 12 rocks to build it. As I already said pack your things and try to find a new place to live in, possibly near a herd of beefalos or a pigmen village -- manure is an even more fundamental resource than rocks, especially if you have many farm plots and/or berry bushes. Put your camp at night away from spiders' nests, and be prepared to defend yourself from the hounds. Run as fast as you can through the swamp to avoid the tentacles. And remember, even if you die, that's not the end. Even if you've spent 10 or 10,000 days in your current world, however frutrating it may be, your death is just another step in the right direction.
*: since the Underground update stones have become a renewable resource through the Earthquakes, but these require you to be in a Cave to be useful. In DST, rocks are renewable thanks to meteorite impacts at rocky biomes.
Since the Underground update, rocks are now renewable in Caves because of Earthquakes. If you're out of Stones or Flint in the overworld, you might want to go into a cave and mine some Stalagmites, wait for an Earthquake, or try to trigger one with an explosive like Gunpowder.
You have no choice but to explore other islands. Pack yourself a backpack, food, and armor. I'd recommend waiting for a hound attack before leaving, and then take off.
As you explore around, be on the lookout for Beeafalos and watch where they occur, so that you can run to them when you hear the sounds of an imminent attack. It's difficult to get hounds to switch targets with other creatures, but possible. Hounds run about the same speed as you.
When you explore for rocks, you don't want to spend your time picking up other things, and instead focus on finding a rock biome as soon as possible. Biomes can be inter-mixed but in general what you're interested in is a large rock biome, so you don't have to explore each nook and cranny of every island. Focus on finding a rock biome, grabbing some rocks, and head home. You can always have more than one base.