Three and a half frames. I hope you have an extra super, brcause if there are only three frames left, they are indeed making preparation for swarming. If the queen can lay 400 eggs/day, how many days do you figure before all the cells have eggs? This does not include the nectar/pollen they are bringing in. The queen is probably already getting thin in preparation for flying, this makes her hard to find. The burr comb usually indicates tight quarters (or Carniolian bees).

First thing I would do is add a deep super for an additional brood chamber. Then, if you find an egg in the queen cell, you can either go for a division or destroy it if they are inhabiting the top super.

It's very hard to raise package bees on undrawn foundation so you are doing quite well. Keep moving undrawn foundation up to but not into the brood nest and as the bees begin to draw it out the queen will place an egg.
Keep feeding until you have at least 2 deeps worth drawn.
Comb that runs wild can be scraped off with a hive tool.
While it is possible you have swarm cells it's far more likely the bees began a few emergency queen cells until the package queen was up to snuff. It's been 12 days here is what to look for. In worker cells it's an egg for 3 days, then it's uncapped larvae for 6 more and then for 12 days it's capped larvae. You just might have a few capped larvae at this stage and a bunch of c shaped larvae.
My first 2 nucs on foundation about 10 years ago did swarm but I couldn't be certain of the queens I recieved ages.
I ended up with 5 double deep hives from climbing trees to retrieve the 3 swarms that did issue!
Right now you really don't have enough bees to make a division your population is falling untill workers emerge in another week or two.
They will take warm syrup up faster than cold too.
Keep their bellies full and keep the frames 10 to a box and tight until drawn.

I want to address the idea of tearing out queen cells.

I don't think it's wise. First of all, I don't think I've seen anything that would indicate that it would thwart the swarming instinct of the bees. Second, I think we need to trust the bees here.

I read just today that having queens in queen cages, even if only for a day or two, affects them negatively. Remember, they're egglaying machines and being isolated and unable to lay is not a good thing. The thing I was reading went on to point out that almost all queens shipped in cages are superceded -or- leave with a swarm - the same year they're installed. It could be the old queen is damaged a bit by the original transport and the workers are intent on replacing her.

If it's supercedure, there's no gain, but if the queen exits with a swarm, you now have two colonies, presuming you catch the swarm. You can hive up the swarm and even combine it back with the original colony if you want to lose that queen.

But on more than one occasion, and I've experienced this myself, the keeper culls all the queen cells, and the old queen leaves with half the workers anyway, and you are now queenless.

Bees have been surviving for millenia without our culling the queen cells, etc. In fact, if it weren't for the mites and diseases that we've introduced or made resistent to treatment, they still wouldn't need us at all.

Leave the queen cells. Tell yourself you are NOT smarter than 10,000 years of genetic imprinting. If they swarm, give them a nice home. If the swarm gets away, wish em well, and pray that they might survive the first winter of low food, and the second winter of mites.

You know, with all the advice we offer here, the best advice we can give other beekeepers is, (1) keep pests and diseases under control AND (2) don't mess around and kill the colony.

On the other hand, I have had good luck with preventing a swarm if there is no egg in the queen cell and more room is added. Advocating a complete 'hands off' approach would also mean not extracting the honey (bees don't make it for us), never re-queening, and never making a division. Laissez-faire with bees would mean that we can go back to skeps. I always assumed that the reasons for movable frames was so that we could manipulate the bees. In the spring, is bottom supering still in vogue? Isn't that messing with the bees' natural tendency for the queen to go to the top? Don't we still use queen excluders to thwart the bees' natural tendancy to mix some honey with brood? The same thing applies to destroying the queen cells.

The queen does not start and stop laying like a hen that is moulting. When conditions force the workers to start supercedure on a new queen, it is usually from lack of space. I would very much like to see the source of the information that says that "almost all queens shipped in cages are superceded". Of all the times I re-queened, there were only a handfull of times where the hive did not immediately settle down and be queen-right for the rest of that year. A couple of times when I was slow to give them extra room and the season was extraordinary they did swarm, but when I examined the queen cells, the little worm was already big enough that the old queen was no longer laying. (I have made divisions at this time if early enough in the season. The hardest part is finding the old queen since she is only a bit larger than a worker.) The progression of supercedure is easily observed, and only a fool would destroy a queen cell with a maggot that spans the bottom of it. However, with the queen only being together for 2 weeks (thanks for pointing that out bamboo), I seriously doubt that the cells are even completely constructed.

If I had a 10 frame deep super with a new queen, spring time conditions, and the start of burr comb and queen cells, I would immediately clean the queen cells, give them another super and wait a couple of days before peeking in again. Most times they will move into the new space and go back to making a strong colony, rather than continuing to go down the swarming route.

And should somebody have an authorative source for the info about the great majority of imported queens absconding before the end of the season, I'm sure many of us here would like to read it.

I have had packages that had 6 frames drawn out and another box of foundation build swarm cells (over 20) and in my lack of knowlage at the time I cut them out and ended up with a queenless hive.

Now "IF" you could find the old queen you can create an "artificial" swarm by takeing her and one frame and puting it in a hive with 9 frames of foundation and leaveing that box in the place where the hive is and moveing the rest of the hive at least 10` away and a turned 90 degrees so now all the field bees go back to the orginal hive and they "think" they have swarmed and the second hive produces a new queen but the "trick" is to find the old gal.

Another one is to "nuc" it out or place frames with swarm cells in nuc boxes and raise several queens.

Now haveing said all this what we read will not always work for us "we" have to find what works for us and that is why I like this hobby so much there are so many ways to "raise" bees!!!