How do I stop chain from moving and jamming up when I stop pedaling my multi-speed bike?
That means that "drive" is coming in through the rear wheel.  

Normally there's a one-way escapement in the rear wheel's hub called a freewheel or a freehub, but they do the same job, which is to let the rear wheel spin when the cassette remains still.

Yours sounds like it has sufficient dirt in it to carry a rotational torque across the interface.  Fancy words saying its dirty, and not disengaging properly.

Inside the freehub will be a mechanism something like this:



Those Pawls (not pauls) push one way, but when your wheel is going faster than the cassette, they're supposed to fold down against some spring pressure.  That's what makes the click or buzz or whirr of coasting.

In your case, they're not fully moving out of the way, or there's another obstruction in there, which is allowing the wheel to push the cassette around.

SOLUTION Open it and clean it. 


Remove rear wheel from bike.
Remove cassette from wheel.

If you have a freewheel, then the clicky parts will be on the cassette.
If you have a freehub, the clicky parts will stay on the wheel (this can be removed from wheel but exact methods vary&mdash;often an 8/10/12 mm hex tool is needed.

Manipulate the parts and see where they're crunchy&mdash;that's the cause of your problem. You can probably use solvents or cleaners to improve the situation, by soaking the part, or pushing the cleaner fluids through cracks. If you have an ultrasonic cleaner then that can be perfect.  Rinse, flush, repeat, dry, and if they're running better, relube and reassemble.


If that doesn't help a lot, you may need to open the freehub.  THIS IS RISKY. There could be a hundred small ball bearings inside this unit, and no guarantee it's user-serviceable.  A straight-up replacement might be your answer instead, but do be aware there are many brands and fitment designs, so you need the same as what you removed.

More info at https://www.missionbicycle.com/how-do-freewheels-work