The game in question is "Scavenge", and it's so sought after by one Amiga fan, that he's gone to the trouble of setting up a blog in an effort to aid the search.
Located at http://scavengeamiga.blogspot.co.uk/, the blog contains background information about this lost title and also details how far the (so far fruitless) search has progressed.
Here's Scavenge blog maintainer, Peter Hull, with further details;
"I have been searching for the full, registered version of Scavenge for many years now but alas can still only find the Demo version.Peter continues;
To have received the full version of the game you would have had to have ordered a copy from the game author, James A. Wright. The full game obviously offers many more game play enhancements".
"James is no longer contactable at his previous ST.Albans address and I have had no luck in tracing him through the usual methods.
Over the years of my search I have posted requests on many Amiga Forums and video gaming sites, I even set up a Facebook page which I have now closed as this proved to just fill my inbox with junk on a daily basis.
I have set up this blog in the hope that someone who maybe knows James or perhaps has the full version of the game can make contact with me. The blog also offers everyone the chance to take a look at the game Scavenge, it is an excellent example of the Shareware and PD games that were available on the Amiga".
Do you have the full registered version of this Amiga Shareware classic? If you do, please contact Peter via the Scavenge blog and put the man out of his misery.