Hopefully cleaning it also saw you check nothing got in there to rub on something to do the playing card in the spokes of your bike effect equivalent.
What you describe with it spinning up and down is on the normal side of behaviour -- spinning constantly would be take a look at thermal paste/pad time as either dried out or a hot spot sending it spinning, though it could still be justified/make a difference.
If you are opening for a case swap then why not find a replacement (for a few years yet they are bound to be around, and when the switch becomes retro then 3d printing and such will probably be more than adequate). Many fans in things these days are all kinds of sealed bearings and whatnot* which means greasing them can be hard/you want to be a watch maker to stand a decent chance of doing much there and for the effort you might as well pay the nice man on ebay or whatever that pulls apart dead ones for parts. Or if you prefer I am not a watch maker but I have many of their skills (like to fix my own precision measuring tools) and will spend five hours rather than $5 to fix something but would still pay out for this, with the only reservation being the thermal paste thing above or that the replacement will have been ripped out of something in if not similar condition then only a few months behind on the deterioration curve.
*15 years ago and still today for house/desk fans (see a dead one on the street and chances are it can be fixed with a drop of oil) you could pull things apart, remove stickers and get a little drop of thin oil into the bearings to have it all come back to life nicely.
