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Llevo is correct - the shocks (in the strut) just dampen the suspension - they do not carry any weight - that's what the springs are for.
when the dampers blow, the car obviously starts to bottom out. if the car is lopsided, you need to consider which side it is on and determine whether or not there is too much weight on that side for that blown damper, compressing the spring itself, and making that particular side lopsided because you are basically sitting on a strut that has absolutely no absorption to the weight other than the springs themselves. you are basically sitting on springs, and not a complete strut. the damper absorbs the weight(energy) of the vehicle and holds the whole strut up to spec. if theres no damper, energy would be stored in the spring-all the weight will be sitting on the spring. you notice people with blown dampers have cars that sit lower, and when in motion bounces like crazy, especially very old cars whose owner could care less about. springs could be a problem, but is least likely the problem. if the spring is sat incorrectly than thats obviously a spring problem. springs RARELY corrode unless its been siding in weather that has a lot of dew (fog) for a very very long time or has been sitting in water or snow which is rarely the case since our springs are up above the car at a considerable amount
try to mount on a high rated (stiff) spring vs. a soft oem spring on a BLOWN damper. you will notice the difference. a lot of coilovers get shot but still retain their height because it can hold so much weight. once the car moves on shot down coilovers, the car sags like dog ****. the op has oem suspension and it is extremely soft. it will sag if the dampers are shot
so responding to the op, your oem springs could be the result of your lopsided vehicle, and the damper is the cause of your lopsided vehicle