Where can I get a new or used balancing ring, part no. 3934565, for a Kenmore 110.42722200 compact washing machine? OR can anyone tell me how much and what liquid should be in the balance ring?

probably not
the machine becomes unbalanced for several reason
with not clothes
the machine is not level
the tub suspension has failed and the tub is not level ( possible your tub ring statement)
if filled with clothes
uneven materials --heavy trousers on one side and light fabrics on the other
have a washer mechanic check out the fault for a proper diagnosis.

if you have a regular top load direct drive washer and if the tub is tilted off balance remove the frame of the washer,look at the back bottom of the frame,there's a spring that is connected to the bottom of the frame and hooks the the back of the tub,sometimes the frame rusts out and the spring rips through the drill hole in the frame,you have to drill a new hole in the frame and reconnect the spring.i've never seen a balance ring damaged.so this is the first thing i would check if washer is off balance.
https://www.youtube.com/watch?v=tbBnr4aCGjc

Its basically saline water so add that.

http://www.askmehelpdesk.com/plumbing/whirlpool-out-balance-spin-cycle-18991.html
http://fixitnow.com/sbrvrs/washers.htm

http://www.repairclinic.com/Washing-Machine-Troubleshooting


This model of washer does not have an out of balance cut off switch. If the load is not balanced properly the tub could hit the cabinet sides. If the unit spins evenly while empty & out of balance while loaded, I recommend checking & balancing the load. If the basket spins out of balance while empty, I recommend having the suspension springs & balance ring checked. Be sure not to wash light fabric with heavy fabric clothes. Also ensure that the adjustable feet are level, that's most important.

See the following Whirlpool Service Manual for their Cabrio washer. The Kenmore Oasis is basically the same machine.

Yes it does have a balancing (actually 2) ring(s).

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/409753510a90c8be852571e90073421b/$FILE/8178583.pdf

Most has a balance ring built onto to the tub, make sure thr balance ring is not damaged, It is normal for the tub to shake in the spin cycle with nothing in washer

Yes, look into trading in the Kenmore for one of only two manufacturers that employ a dynamic balance system built into the washer drum. Samsung and LG both use this dynamic balance system that balances the load while it is in motion. They also use a vibration sensing system to modify the time and speed of the spin cycles to minimize any vibration that might still be present. LG has this available on only their upper level models while Samsung's VRT is available on most all models. I believe that both of these manufacturer's are available from Sears.

to cause unit to go off balance taking in fact that you put clothes in washer even...check your counter balance spring it hold the tub to base at rear of unit,,,and check your balance ring at the top of your wash basket its fill with water if cracked will cause this problem,,

check to see on the top of the basket if there is a balance ring it will look like a large doughnut if it has one and if it is loose or if the fluid in the ring has leaked out then this will cause the out of balance while in spin let me know if i can help 276 523 4434

The only thing I can suggest is to make sure the load is evenly distributed around the circumference of the tub. If you do only light to medium loads, this can occur from time to time. Lift lid as the unit goes into the spin cycle to check. If it is evenly distributed, you may have a support problem.