Both of these are sometimes itchy conditions. We had a case in which it took three attempts to biopsy, taking two or three samples each time, to find the Demodex organisms in a group of infected cats. Some cats have eosinophilic granuloma complex that resembles feline acne but these are supposed to be distinguishable by examination of biopsy samples. Eosinophilic granuloma complex can be caused by allergies and flea bite hypersensitivity. If the new cat brought fleas with it, this might be a consideration.
Feline acne is usually treatable by topical treatment only.
When topical treatment alone is not sufficient, the use of systemic antibiotics or corticosteroids is sometimes necessary. Most cats will respond to corticosteroid treatment within a couple of weeks. If you have not contacted the dermatologist to report that treatment does not seem to have been successful, that would be a good thing to do. The dermatologist may suggest further testing, especially with the apparent contagiousness and the persistence of this problem or she may feel that it is reasonable to try some of the other treatments to see if they are more effective. Demodecosis in cats is usually responsive to lime sulfur dips, at least according to the textbooks.