Do they really hand out severance when you're voluntarily leaving? Typically that's for layoffs or "soft" firings.

For cases when you sign a noncompete when you leave, that's going to be in the case if a layoff or soft firing where they've got leverage over you. Otherwise fron what I've seen on NY where noncompete can be enforced, the noncompete is signed when you get the job. For the case where you're signing things when you leave though, that's when they've got a severance to hold over your head.