Driving LED arrays
1 - The lamp I have has 24 LEDs all in parallel, with no resistors at all and is driven by 4AA cells. The current distribution is inevitably very uneven and some LEDs are much brighter than others. As others have said, a bad situation relying on the cells' internal resistance (and possibly, that with 24 LEDs, it won't matter if a few burn out and a few are too under-driven to light up!) We really should have individual resistors, but as you say your arrays of 9 seem happy enough to run from a 4.5V adapter, let's go on from there.
2 - If one array of 9 is running at 4.5V and drawing 20mA per LED, that is 4.5V x 20mA x9 = 810mW. (Personally I'd guess they are running at a lower Voltage, say 3.5 - 4V, and I'd be happier if the average current were a bit lower, say 15mA, to allow for the uneven split between LEDs.* That would work out nearer 500mW.) Therefore, if you want 8 of these arrays, you need 8 x 810mW of power = 6.48W (or for me 4W). This you can get from either of your power supplies, 12V x 3.7A = 44W or 16V x 1A = 16W. And in each case there is plenty of power to spare for dissipation in some resistors.
3 - Now, how should we wire them up? Each module needs 4.5V and 180mA.
So the 12V supply can have up to 2 in series (2 x 4.5 V < 12V, but 3x4.5V >12V)
and it can have up to 20 in parallel (20 x180mA < 3.7A)
The 16 V supply can have up to 3 in series (3 x 4.5V < 16V)
and up to 5 in parallel (5 x 180mA < 1A)
So 2 in series x 4 in parallel will work on both.
This will need 2 x 4.5V = 9V and 4 x 180mA = 720mA.
4 - So for the 12V supply we need to drop 3V. This is best done for each of the 4 parallel circuits separately. So the resistor for each chain is 3V / 0.18A = 14.4 or say 15 Ohm.
For the 16V supply we need to drop 7V and the resistor for each chain is 7V / 0.18A = 39 Ohm.
Of these two, I would prefer the latter because the greater resistance gives a little more current limiting stability. It does however require that the resistor dissipate 7V x 0.18A = 1.26W, so at least a 2W resistor is needed.
5 - If you use the 12V supply, the best option might be to run 8 modules in parallel with a resistor in series with each module. This would require dropping 12V - 4.5V = 7.5V in the resistors. Each of the 8 resistors would be 7.5V / 0.18A = 41.6 or say 43 Ohm. This would dissipate 7.5V x 0.18A = 1.35 W, again at least a 2W resistor.
6 - And as I said I'd like to reduce the average current to say 15mA per LED = 135mA per module. This gives a resistor around 60 Ohm, say 62 Ohm which sets the average current per LED to 13.5 to 15.5mA (Sorry, it's getting late so I've skipped the calcs!)
Apologies for no diagram: if wanted, I'll scan one in tomorrow and post it.
* If 9 LEDs in parallel draw an average of 20mA, some will probably get more of the current than others. Those that get less will just be dimmer, but those that get more are likely to be getting overheated and shortening their lives. Due to the characteristics of LEDs, most will draw a little less than average and a few will draw a lot more: so the average needs to be well below the safe maximum. All of the above answers fail to deal with this problem: the only complete solution requires individual resistors for each LED.