Here's a handy list of the current known resources and their availability:

Renewable

• Grass
• Twigs
• Reeds
• Berries: infinitely renewable in the wild. If dug up and replanted, renewable but must be fertilized after every five berry harvests.
• Manure: infinitely available, as long as you have a herd of beefalo near you, or a pigmen village (feed them vegetables or flower petals), or a koalefant.
• Beefalos: renewable if a herd exists. Calves are born every autumn which add to the herd. Kill the last beefalo and it becomes extinct.
• Pigmen: generated indefinitely from their village or from pig houses. If the pig in a house is killed, the house generates a new pig in four days.
• Wood: renewable. Trees can be planted back from pine cones, unless you destroy the pine cones or burn the forest. Pro tip: dig up the tree stump to get one more log.
• Gold: renewable. Gold from gold-veined boulders don't respawn, but if you find a Pig King you can feed him meat (a renewable resource) or eggs (feed a bird in a cage monster meat to get them, or steal them from pengulls) or give him trinkets to get gold nuggets in return. Cave quakes can drop gold.
• Flowers: renewable. Flowers spawn butterflies which, when caught with a bug net and planted, yield more flowers
• Spiders: putting them in the resources because of the Silk. The nests generate new spiders when you kill them, up to their current limit (it should be 6 for the small nest, 9 for the medium and 12 for the big). If you kill all the spiders in a nest, they'll regenerate over time. If you destroy a tier 1 or 2 spider nest, you get spider silk but the nest is gone for good. But if you destroy a tier 3 spider nest (it has three lumps of web), you get six spider silk and one spider egg that can be planted to create a new spider nest. (The only way to increase the number of spider nests is to let a tier 3 nest become a Spider Queen; she leaves behind a tier 1 nest, walks around for a day or two, and then becomes another tier 1 nest.)
• Rabbits: they're regenerated as long as their holes are intact; if you dig up the hole, a rabbit comes out, but the hole is permanently destroyed
• Turkeys: they're vermin and they're there to gobble on your berries! Randomly generated in any bush, you don't have to worry about their quantity.
• Flint: Those that you pick up don't respawn anywhere else on the surface. They drop in caves on earthquakes;
• Rocks: Destroying a boulder or a stalagmite yields rocks, flint and occasionally gold, but boulders don't respawn. They drop in caves on earthquakes;
• Frogs: continuously generated from ponds. Frog Legs are also left when a Merm is killed.
• Fish: continuously generated from ponds. Fish are also left behind when a Merm is killed.
• Bees: continuously generated by beehives or bee boxes, up to a maximum of four per hive/box.
• All Farmable crops: your farms keep generating crops, needing the occasional (renewable) manure to fertilise the farm.
• Honey: You can get honey by killing bees (the bees will renew); or by creating bee boxes and then harvesting them periodically. (You can also get honey from destroying a beehive, but beehives don't appear to be renewable.)
• Mushrooms: They come back in a few days as long as you pick them. If you dig one, you'll get two at once but they'll never appear there again.

Not renewable

• Saplings - once they're gone (e.g. by being burned), they're gone. You can replant them anywhere, though.
• Berry Bushes - likewise. If you dig up a berry bush, you'll need to fertilize it when you plant it, and again periodically, to keep it producing berries.
• Grass Tufts - likewise. If you dig up a grass tuft, you'll need to fertilize it when you plant it, and again periodically, to keep it producing grass.
• Rabbit Holes - if you dig one up, it's gone forever
• Mushroom locations - if you dig one up, it's gone forever
• Honeycomb - you get one if you destroy a beehive, and you can use it to create a bee box. But it appears that the number of beehives you start with is all you get. You can also get honeycombs from killer bee hives.

As you can see, the list is still a work in progress, since I'm trying to discover everything on my own. I'll try to keep this updated every time I find something new. Keep in mind anyway that new features are constantly added to Don't Starve, so these information is not fixed and still likely to be changed from time to time.

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.