Hi there
From the sounds of things in your question, yes, you will likely be better off targeting your internal pages to rank for relevant search terms rather than your homepage as a 'catch-all'.
So e.g. http://xxxxxx.org/the-keyword-for-ranking try to acquire (natural, not paid) inbound links to these pages from relevant authoritive websites with relevant anchor text.
Your homepage will acquire links anyway (usual scenario), this page will rank for your brand and perhaps main overall keyword, difficult to tell though without knowing anything about the website.
There was an interesting White Board Friday recently entitled "Mapping Keywords to Content for Maximum Impact" which is worth a view. It introduces the concept of the 'Frankenpage' which is best avoided (e.g. a homepage or internal page that tries to cater for too many different topics and keywords), better off targeting 1 page for one or two phrases based around 1 main topic.
Hope that helps,
Regards
Simon