Sometimes (actually more often than not, but NOT by any means ALL the time) when a cat is placed in a new environment, they will pick up little bugs or germs not present in their previous environment. Most will get through it with a little sneezing and runny nose/eyes for a short while, but normally will get through it by themselves without any medical intervention. It has been my experience that unless it is a 24/7 symptom, and it is obviously an infection or virus (green mucous = bacterial infection and WILL respond to antibiotics, yellow or clear mucous = viral and most probably WON'T respond to antibiotics) a trip to the vet is not warranted. Give it a few more days and if it still concerns you, then maybe place a call to the vet, but I wouldn't allow a course of antibiotics unless you KNOW it is an infection. (Have the vet do a culture before he prescribes) While the over- and mis-use of antibiotics is a personal soap box issue for me, it is one that bears discussion. We weaken the immune system's response when we give antibiotics when they are not called for. It just isn't a good idea.
OK, that now being said - drooling is another matter. Drooling can be a sign of many things - among them happiness and contentment - but it can also be a sign of bad teeth/gums. Do you also notice a bad odor to her breath? If so, then have her teeth checked out. Drooling can also be an early symptom of chronic renal failure, but there would be other symptoms you would notice, such as excessive water intake/urine output, a hunched-up appearance to the back area, and maybe some difficulties in walking or moving around normally.
Hope this helps you,