The Snail Puzzle

I have a nice puzzle book called "Mega Puzzles" from Hinkler books. While there are many interesting puzzles there, I have been concentrating on one for the past few days. It's called the snail puzzle. Below is a picture of the first of the two puzzles that appear on page 72 as puzzle 72:

The spiraled letters compound our difficulties so we can straighten out the letters:

A O E M U A R Z O P N T I N S D H H I N A E A L R E R M O L I C E N I A O N T S E G E S

Looking at the puzzle "AMAZON" seemed an obvious guess more or less at once.
The first letter is 'A' and "M..A..'Z' is there to.  It fits. Actually it can fit in a few more ways.
(Note "AAR" would fit too, but there are 47 letters and 7 rivers for slightly more than 6 characters per river name on average.)
 

A O E M U A R Z O P N T I N S D H H I N A E A L R E R M O L I C E N I A O N T S E G E S

A        M      A    Z O    N

 We also need one river ending in 'S'. The ' E ..U" in the front suggested "EUPHRATES" which would give the final 'S'.. It fits to - and seems too long to be a coincidence.

A O E M U A R Z O P N T I N S D H H I N A E A L R E R M O L I C E N I A O N T S E G E S
A        M      A    Z O    N
        E      U               P                      H                          R                                    A        T     E        S

  As before, the first matches that come to mind are shown and we can adjust them later

Now "ORINOCO" practically jumps out:

A O E M U A R Z O P N T I N S D H H I N A E A L R E R M O L I C E N I A O N T S E G E S
A        M      A    Z O    N
        E      U               P                      H                          R                                    A        T     E        S
   O                  R                   I                      N                               O       C               O

Four more to go- maybe. "OHIO" might also have jumped out to me.

And the unused "T" - what river begins with "'T'?
TIBER?             (No 'B'..)
THAMES?       (Fits.)
Something I have not thought of?

Well, if I had any genius for this type of puzzle, I suppose I could solve it in a few more minutes but I puzzled for an hour - so I gave up. But, lo and behold, the solution in the book for number 72 does not match the puzzle in any way! I checked around for something out of order but still could not find any suitable answers. So I fell back on ingenuity - on research and on rule programming!

The story of that process is best left to a page. So you can see the page SnailPuzzle if you want to see the rather arduous method I employed to come up with a solution.