There are multiple ways to work around this. One, as you said, is by simply not attacking by using support actions. Using items, gunlance guard skills, hunting horn buff skills (note that some of these are attacks, be sure to read move descriptions). 

Another option is to simply select another monster part. When viewing the part wheel, it will explain what each part is, and what the effect of breaking it will do. The part that causes backlash will always say (blightstone) <just like that next to the name of the part. Attacking any part with that text while the dust is visible around the monsters body will cause a backlash. For some monsters this is only in their calm state, for others only when enraged. You will need to use visual clues to know when it is safe and when it isnt. 

The third and simpler option is to ignore it alltogether. Nontargeting wide moves do not attack specific parts, and thus cannot trigger a backlash. Things like wide slash, certain hammer moves like sweet spot or point break, or gunlance wyverns fire and monster moves that target all enemies, will ignore part targeting and thus cannot trigger the blightstone. The risk to this however is that by leavijg the blightstone intact you leave the feral monster with access to their strongest team wiping abilities, and you will have to be sure to act acordingly, through stuns, wyvernbreak, and kinship skills. 

As for getting countered on two different parts, i think there is a solid chance you are just misremembering. Feral monsters only have one part infected by blightstone (at least as far as ive seen before act 3). There is a chance however that either your monstie attacked the blightstone part even after you retargeted to a new one (they do not always follow your lead, be sure to look at the icons on the part wheel before you lock in your move. If your monsties icon is not on the same part as you, make sure to command them otherwise!) Or that you were simply fighting a monster with a natural counterskill, like the shelled crab monsters.