I heard that we should have separate keywords for each header. But Every site that I checked that was ahead of me has the same keywords for each page? Any thoughts?

Joe,
Just because a website outranks you does not mean that they are doing everything right. A lot of the time websites can rank in a market based off of doing just a few things right - for instance they could have been doing a lot of link exchanges over the years and the authority of this alone could place them in a positition that helps them outrank your site.

This being said, you can take advantage of proper copywriting, including unique headers, title, descriptions and content - in order to get the "things right" that they have not taken advantage of. This will not help you outrank them right away, especially in Google, as it is so link centric, however it will help you overtake them as you approach their level of pagerank and authority. Also, you will capture many more keywords than they will as your optimization will be more focused, and targeted.
End result - more traffic, more leads, more sales.

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I heard that we should have separate keywords for each header.

As well as unique meta tags and content for each page. I'm continually surprised at the variety of search strings that drive traffic to my customers' sites.

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Looks like a got a lot of work to do.