You can indeed make use of SAM to delete the achievements, the reason why playing on different pcs pushes you back to the previous state of achievements would be because you have data saved there that is different from the data you have currently.
This behavior can be also seen with games that have savefiles backed up on the cloud:
In that case you will be asked if you want to keep the local or cloud save.
Assuming you can indeed succeessfully modify your achievments trough SAM you will be able to play on other pcs assuming those do not contain data that would instantly unlock said achievements.
EDIT: As long as you play on a computer that has data with achievement unlocked it will re-unlock these. this is also valid for savefiles! therefore, delete these or back them up elsewhere. in case your cloud data stored stats that the game checks to unlock these achievements, you can decide to go offline make this change and when prompted upload local files to cloud and overwrite old files. last option is if these checks are doing serverside (which i have yet to encounter but i've heard people mention this possibility before) in which case you might be out of luck, altho you could at least try using the stat tab in SAM and for example for an achievment stating
"have at least 1k hours of playtime"
if that stat is currecntly 2k, you can lock the cahivment but since the playtime will be over 1k as soon as you open it it will re-unlock.
if you follow all of these steps correctly you should be able to relock all your achievements without issues unless they are stored server side and updated in a sensible manner for example not allowing lower values than previously recorded (which i doubt)
fun-fact: This is how, for example, from-soft checks your inventory in DS3, in this case for added stuff you are not supposed to have access to yet, not for stuff being taken away.
NOTE: not all games support steam family share, ARMA3 is one of such games, therefore the family share answer above is, in my opinion, incorrect or at the very least situational.