I often read the phrase "not to" preceding an action, as in "not to run" or "not to swim". It seems awkward. Please explain explain the usage.
More than one usage of 'to' exists. Restricting analysis to to + infinitive, note the difference between
Not to be charged the full price would be great.
To not be charged the full price, make sure you show your concessionary pass.
The second example here uses the 'in order [not] to' sense. Possibly, positioning has evolved to show the distinction where needed.
I think the negative to-infinitive is normally not to do ( do stands for any verb) as used by Hamlet:
to be or not to be
I don't see anything awkward as to this form.
Occasionally you find an alternative form: to not do. But there is no difference. Of course, there may be sentences where the latter form is justfied stylistically.