Cheap Soldering Iron vs Cheap Solder
When you say "cheap soldering irons", you ain't just whistling Dixie. If this happens with rosin-core solder (not acid-core) then your problem is that the tips are not iron-plated. And frankly, I didn't think they made them like that any more.

If you can get a decent tip, this should solve the problem, but ensuring that you've got proper materials is not likely to be easy.

Since you've apparently been using these irons for a while, it seems that you'll be soldering for a while longer. I'd recommend trashing both of them and get a good temperature-controlled iron. And you can keep on using cheap solder, as long as it's not acid-core.