(A very humble attempt to answer my own question. Feel free to downvote if I'm wrong. I'm not even a native speaker.)
"Try to avoid" is very close to "Try to not". We use these wordings when we mean to dodge something, literally or figuratively. ("To dodge a bullet" = "To avoid a bullet".)
Example 1
- Try not to kill him.
- Try to not kill him.
- Try to avoid killing him.
1 could either mean that
- Option 1: Your aim is something other than killing him. (You don't care much whether or not he will be killed.) - or -
- ... Option 2: Your aim is keeping him alive. (He is your friend, you don't want him to be killed.)
2 and 3 definitely means the first option.
Example 2
The same goes for "Try not to use it", "Try to not use it", and "Try to avoid it".
- This is a big red button that launches nuclear weapons. Try not to use it.
- This is a big red button that launches nuclear weapons. Try to not use it.
- This is a big red button that launches nuclear weapons. Try to avoid using it.
1 could either mean that
- Option 1: You don't care much whether or not the new world war will start. - or -
- ... Option 2: You don't want to start the new world war.
2 and 3 definitely means the first option.
"Try to avoid using it" vs. "Try to avoid it"
To my opinion, the former one can always be replaced with the latter without any difference in the meaning. The only difference is tone: the latter one is less formal.
- Avoid using violent language. (A rule in the corporate style guide for in-house writers.)
- Dude, please, avoid violent language. Seriously.
An addendum
The choice can also be dictated by considerations of aesthetics. As John Lawler said:
"Try not to spill the coffee", for instance, is simply the idiomatic way to warn someone; "try to not spill the coffee" is unidiomatic and therefore raises the question of why the idiom was not used. One answer might be that in the previous discourse the phrase to not spill (versus to spill) had been used, and was repeated in this sentence.
For a straight comparison, you would need "Try to avoid using it…"
Failing that, "Try not to use it…" constrains your own actions; "Try to avoid it…" limits your prescribed response to other people's actions.
– Robbie Goodwin Jun 26 '20 at 22:11