HI there, this is Dr. Elaine
Brownish (or other colored) liquid material dripping from the mouth could be from the respiratory tract (airways). Sometimes it is from the crop (digestive tract) and the bird is actually regurgitating. Most commonly, though the airways are the source, and usually due to infection, sometimes parasites.
Respiratory infection is not uncommon in pet chickens, and is often initiated by a virus. Some birds will respond to antibiotics IF bacteria is involved, and IF that particular antibiotic is effective against it. Antibiotic resistance is a huge problem in medicine today, both human and animal. Indiscriminate use of antibiotics is a big reason for this.
Supportive care for an ill bird such as her involves (like you have started) warmth, isolation (for her protection and that of the flock). Nursing care--i.e. keep enclosure clean/dry. Entice to eat with soft, cool, moist foods if not eating well (i.e. melon bits, cooked squash, soaked grain bread, fruits, soaked or cooked cereals like Cheerios).
Aspirin in the water is sometimes used to help increase the bird's comfort. Use 5 regular strength tablets per gallon of drinking water, or 1 and 1/4 per quart.
Having her seen may provide a much better chance of having her treated more accurately, especially when it comes to potential drug/antibiotic selection. Diagnosis also helps in terms of how to protect the rest of the flock and what to expect.