I used WD40 and very fine sandpaper to remove the rust (didn't have Scotchbrite pads). Still a few dark spots, but happy it wasn't worse.
For recent onset surface rust on things (I don't have an ax to be specific about that tool), I have found that those Bounce Dryer Sheets (antistatic things) are good for cleaning it off. They don't rip, and have enough toughness to the fibers to make very good super-mild abrasive wipes. We use discarded dryer sheets to clean many things - bathroom sinks, tubs, mud/dirt on shoes, the stovetop, etc. They are some of the best cleaning rags around - and they're free if you would have instead discarded them after using them in your dryer. They are not absorbent, but they are good scrubbers that you can follow up with a paper towel if needed.
They're also handy if you have longer hair and live in a cooler/dryer climate to get the static out of your hair (I use a fresh, clean one for that task however!)
If you reload ammo, a dryer sheet is a great addition to your brass tumbler to clean things out. I made the mistake once of ordering tumbling media that had some red rouge mixed in with it. That turned out to be a royal mess. Dryer sheet to the rescue for cleanup!