The one and ONLY type of solder that should be used on electrical connections is rosin core solder. There is absolutely no need to use a separate flux!!
I have worked in the electronics repair industry for decades. I have made tens of thousands of solder joints. I have never, ever, used any type of flux paste or liquid for any of those solder joints. The rosin-based flux is in the core of the solder! That’s why it’s called “rosin core solder.”
I agree with others here that this is not rust. It’s just old, excess flux. If you were to heat these joints, that stuff would bubble up and evaporate. If you don’t use any flux and just rely on the flux in the solder, you won’t have this problem.
