My calico cat sneezes multiple times a day, releasing mucus. This began about three months ago. She is also audibly congested. Does she have an allergy? A cold that won't go away? Perhaps this seems obvious, but do I need to take her to a vet? She's very skittish.
Yes, she needs to see a vet to be properly diagnosed.
One way to make vet visits easier on shy cats is to use a mobile vet who will come to your home. Then you only have to deal with stranger and handling stress, and not the added stresses of a carrier, travel, and a strange place.
Another way to ease the stress of a vet visit is to look for a Cat Friendly Vet. This designation indicates that the vet has additional training in easing cat stress at appointments. This can include things like "incorporated a waiting room/area that reduces stress associated with noise, other pets or unfamiliar smells (methods can include feline-only area, cat-only appointment times, going directly into the exam room, etc.)."