Are nudibranchs slimy?

Yes, just like their land-dwelling cousins, nudibranchs definitely leave a slime trail.  This slime trail is actually incredibly useful as a form of communication.  Depending on the situation the nudibranch can excrete various types of chemicals into the slime that can send a message to other nudibranchs.  For instance, they can warn other nudibranchs of impending danger when they are being preyed upon and they can leave a trail for a potential mate to follow.