SnowFella is right about shooting at higher speeds. Many more advanced cameras also have high frame rates available. i.e. the high-end Canons can often do HD in 120fps and the Sony FS7 can even hit 180fps, which looks like a dream when played back at 24fps (extreme slow motion).
One solution is to rent a global shutter camera from a place like lensprotogo or lensrentals. That's of course not free but it beats having to purchase i.e. an Ursa Mini 4K, which is one of the more affordable cameras that would make you happy. Again I would recommend the Sony FS7 if you are willing to figure out how it works (takes a day or two).
Finally, you could make the most of your rolling shutter by setting your shots up carefully. Experiment with the direction of movement in front of the camera. Left to right will be most visibly distorted but what would lessen the effect?