I have some white, fluffy-looking stuff growing on my oleander plant where a branch comes out from another branch. What is it?
What you are describing could be either a scale insect or mealy bug. Both can look like cotton, and when you press it hard with your fingernail you get a brown or reddish liquid. If you don't have too many affected plants, you can just rub the scale off as it appears. Don't let the insect numbers build because they can literally suck the plant to death (these are piercing, sucking insects). If you have large numbers of plants, apply a dormant oil spray.
You should contact your local Extension office and inquire about its diagnostic services. Once the insects are correctly identified, you can form a plan of action.