First you should address how search engines crawl your site and avoid duplicate content issues by making sure you have the www subdomain (www.example.com) redirecting to your root domain (example.com).
The most efficient way to do this is to create a DNS record - see the first example here on how. You should also use your web server's configuration to redirect the www subdomain to the root domain.
After you're sure you have the above redirection working (DNS changes might take up to 72 hours to propagate), the next thing you should do is to specify your preferred domain as the root domain in Google Webmaster Tools under Site Settings -> Preferred domain, as indicated here.
After that, Google will follow any links with www in them as http://example.com, and will display them accordingly as well. Note that it may take some time for pages to reflect the preferred domain update.
Given the above, the answers to your questions are then:
I guess what I'm trying to ask is if I should create another website
profile without the "www" included in the url. Is Google viewing both
versions as two different websites? Am I hurting myself by not using
the www?
No, Google views the www subdomain as part of the same site as the root domain. See this for more on that.
How do they handle adding the verification file when both versions are
really the same site?
Once you have the www subdomain redirecting to the root domain, you likely will not have to verify the root domain, as stated here:
Generally, once you have verified one version of the domain, we can
easily verify the other using the original verification method.
However, if you've removed the file, meta tag, or DNS record, you'll
need to repeat the verification steps.
(Source above from Google)
Also what is weird is that I have a sitemap created for my site, but
webmaster tools says that there isn't one available.
Once you have the above redirection and preferred domain set up, submit your sitemap using Google Webmaster tools as covered here.
