You have been told wrong.
Haste and mastery are the 2 best stats, but that doesn't mean that the others are completely useless.
In fact, the difference between vers and mastery is only about 10 %, so is the difference between crit and vers.
Critical strike chance is the weakest secondary stat for fury, because it does nothing during Battle Cry, which is where we put out a lot of our damage, but being the weakest doesn't mean useless. In a hypothetical situation where you can choose 100 haste or 200 crit, the crit would be better.
Items with significantly higher item level will very often end up being upgrades, even if they have bad stats. The extra strength as well as the amount of stats will outweigh the smaller amount of "better" stat.
Now as for Pawn. It does nothing more or less than take each stat, multiply its amount by the provided value of the stat and adding them together.
It does this for both items and tells you how much bigger the number is for the new item compared to the old.
If you are only interested in maximizing haste and mastery, just put in the stat weights:
haste - 1
mastery - 1
crit - 0
vers - 0
strength - 0
If you are interested in maximizing you dps, you should sim your character and provide Pawn with your personal stat weights.
Example for my alt warrior:
This is what Pawn will do (taking my stats from above, since I don't know yours):
Ring 1: 1960 * 32.23 + 1158 * 29.13 = 96,903
Ring 2: 1469 * 24.13 + 1745 * 32.23 = 91,688
Ring 3: 2725 * 24.13 + 1248 * 26.59 = 98,939
As you can see, ring 3 has so much stats (because of its item level), that it ends up being better (in my case by about 9 % compared to ring 2, it might easily be 14 % in yours, I have pretty badly itemized gear).
There are 2 problems that you have to keep in mind with Pawn:
1. Your stat weights change. That's why you have to use your own rather than depend on the defaults.
It also means that swapping an item might change your stat weights a bit and with the new stat weights the old item will suddenly seem better.
You should keep this in mind and use Pawn as a ballpark. If it says that an item is 10 % upgrade, you can be sure that it is decently better. If it says that an item is 0.2 % increase, you should sim both options, because the stat value fluctuations might give you slightly incorrect results and such a small difference isn't conclusive.
2. Special effect aren't taken into effect. Pawn only looks at the stats and multiplies them by their value.
It is therefore sadly completely useless for comparing non-statstick trinkets, tier items, legendaries, or in your case, the Melandrus ring.
If you have any items with effect that aren't flat stats, you will have to sim those manually.