[verb ない → なくて] + 済む
The grammar structure なくて済む expresses that something undesirable or burdensome doesn't need to happen – that one can get by without doing it. It conveys a sense of relief or fortunate avoidance: the situation turned out (or will turn out) in a way that spares someone from having to do something.
It is formed by attaching なくて (the て-form of the negative ない) to the verb すむ済む (to be settled / to be sufficient). Literally, it means "it is settled without doing ~", i.e., things work out without needing to do the action.
Unlike simply saying 〜~なくてもいい ("you don't have to"), なくて済む carries a stronger nuance of relief or being spared from something unpleasant, troublesome, or costly. It implies that the speaker views the avoided action as a burden.
なくて済むように – "so that one won't have to ~"
なくて済むなら – "if I can get away without doing ~"
Since すむ済む is a regular う-verb, it conjugates freely: