To expand on what Bruce and everyone else is saying, it's amazing how much un-noticed ' black gunk' can build up on a tenon, and more importantly inside the socket. You might think it's acting as a lubricant, but it promotes stickiness. Make sure those are perfectly clean and de-greased before you do anything.
Then, and only then use a little lube if you must - you can get small tubes of 'fishing reel' grease if you need something portable - cork grease isn't really suitable, it tends to stay sticky and attract dust etc.
Talking of which, an old trick with fishing rods is to rub the joint tenon on the back of the hair, picks up just enough lube for the joint to not stick. Back to saxophones, just make sure the tenon are socket are REALLY squeaky clean each time, before you add any lube to the equation. If it still sticks, then it's 'tech time'... (unless you really know what you're doing with a little lapping compound !)