For the most part, Charles, it is a matter of style. Both are correct here, but in your examples, I would definitely prefer 'avoid'. *Avoiding* having to do something is more active and "powerful" (and perfect for expressing an advantage) than simply "not having to do it", which is rather passive and dull. The thing being avoided need not be explicitly unpleasant, it can simply be something I (or the target audience) would prefer to not have to do, or something that I find burdensome/annoying.

See e.g. http://www.ntnu.no/gemini/2002-06e/44.htm
Doctors can speak directly into their PDAs and thus avoid having to use pen and paper on the scene of the catastrophe.

http://en.wikipedia.org/wiki/Wikipedia:Adding_a_GFDL_license...
It means that people who cannot afford copyrighted material can have access to it. For example a third world country could legally include some of your work in lesson plans for its schools and thus avoid having to pay for copyrighted material.

http://www.interventionmag.com/cms/modules.php?op=modload&na...
Maybe it's traditional GOP thinking that since restaurants will have the opportunity to sell expensive food and drink, hotels their accommodations, and so on because they brought their convention to town, the free market is at work so in the end everyone must be happy. Or then again, because they plan to park cruise ships in the harbor, maybe it's a slap in New York's face. They'll bring their own food and drink and thus avoid having to intermingle in the multiculturalism New York is so famous for.

http://web.mit.edu/answers/dialup/dialup_passwordless.html
How can I avoid having to type my PASSWORD each time?

And the list goes on. Good luck.

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Note added at 1 hr 26 mins (2006-01-05 23:56:35 GMT)
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P.S. I agree with JC's remark regarding "one"

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Note added at 1 day 20 hrs 23 mins (2006-01-07 18:53:02 GMT)
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As I understand your question, Charles, you are referring to this one specific type of construction, not the difference between "to not have to" and "to avoid". As a native speaker of English, you obviously know the difference between the expressions in other contexts. However, your examples do not involve simply "to avoid", but rather "to avoid *having to*", which is a different kettle of fish. I think your examples could definitely use some reworking, but as for the point you are trying to get across -- namely that something is creating an advantage -- I would use "avoid [the disadvantageous aspect]" (or some other strong verb) in all of them.

Just as an example:
[Our new and improved packaging method] avoids (or, yes, obviates) the need to open and throw away the wrapping

or

[By using this new method] I avoided having to touch the actual product