small insects... could they have been root mealies? I wouldn't do more than you have, or you'd make the plant poisonous... if it continues to brown, and it's not from being indoors in central air, you'd better just toss it, there are a lot of rosemarys out there.

If you can find some diazinon you can use it as a soil drench. It is not absorbed into the plant and breaks down in soil. Or use pyrethrin as a soil drench.
Another approach would be to wash the roots totally free of soil and repot. You could have fungus gnats - generally not a major problem.

I hate to be other than positive to a new rosemary grower, but you might have more problems than the bugs. Rosemary has a number of needs that are definitely not met in these little "Christmas trees" that are sold at this time of year.
Your plant's interior browning is probably caused by lack of air circulation and resultant fungus - almost inevitable given the fact that there are multiple plants in what I guess is one 6-8" pot. If you still have it in the decorative sleeve, that will also hold water around the plant, at the base and around the lower leaves.
Try to keep it cool (a bit too cool for humans, to be honest - 50F would be just fine) and sunny, give it some gritty soil to replace what you scraped off, and let the soil dry as far as possible without the plant actually wilting, before watering it again. When you water, soak it, then let all the excess drain out before returning it to its sleeve or cache-pot. Don't worry about the bugs. They sound like fungus gnats, and if you don't let the soil surface stay damp, they'll die off. Misting is not usually a good thing, in my opinion, though a trip through the shower helps in midwinter to clean the leaves.
If you manage to carry it successfully through the holiday season, then I'd suggest repotting it as single plants, each with its own pot (probably a 4" square would be about right). Those multi-pots are cheap because it's easy to stuff cuttings in a pot and just shear a mass of it to shape. You can get a better tree, and healthier, for next year by starting early and training your own as a single plant. (Besides, it's more fun...)

How did you make out with your buggy rosemary? I hope you didn't give up on it. I've been growing rosemary indoors (at 70 degrees) for many years and have found that if they are kept moist but not standing in water, they make fine houseplants. When you said you had a tree rosemary was it shaped into a pyramid or was it on one stem like a standard? In any case it was probably forced for sale and would need a period of time to become acclimated.

After years of working in the tropical plants field I would definitely agree with GardenPaws that your rosemary has fugus gnats. They cannot survive if the plant is allowed to dry out more than it has obviously has been kept. They are often an indication that the soil is staying too wet, and a root rot problem often ensues if a more appropriate watering practice isn't adopted.
I would suggest that a clay pot may be of help. Fungus gnats seem to flourish in the anaerobic enviroment that plastic containers tend to create. Plastic does not 'breath" as clay does.
Better air circulaion in the room the plant is in will help, too, not a wind, mind you, but avoid stagnant air. I used to run a small desk fan on low to provide better air circulation where I kept my orchids, when I used to grow them. It doesn't have to blow on the plants but, just keep the air from sitting where they are at.
Fungus gnats are a sign of an anaerobic enviroment. Fix the cause and they will go away.