Hurricane is largely regarded as one of the most powerful weapons in the game, but it has a special category of recoil all its own that needs to be accounted for.
It actually isn't inherently better at stripping shields. This has to do with the fact that shields/barriers do not react differently to rate of fire (in spite of what the tooltips say.) Shield Break will occur at a specific damage total regardless of the weapon type used to do the damage. The advantage that it does have is its DPS, which can certainly increase the speed at which it strips shields.
The key defense to be concerned with when using the Hurricane is armour. Without any armour mitigation, the Hurricane will perform very poorly against armoured targets, even with its impressive DPS. You'll want to use an AP mod on the weapon, or use Drill/AP ammo. Warp ammo will also help, but will not provide any penetration. With mitigation in play, the Hurricane becomes very effective against all enemy types (with the possible exception of the Phantom.)
The Hurricane, in addition to its recoil, also suffers from huge accuracy problems. This creates a very wide "spread" to the bullets. Think of it as a cone - a large target entirely within the cone will take the full damage, but a smaller target at middle distance will only be taking a fraction of the damage, because most of the "bullets" are missing the target due to inaccuracy.
Stabilizing the Hurricane offsets some of the accuracy problems, but the spread will still be an issue at moderate and long distances. With enough bonuses, this can be overcome by the sheer damage output of the weapon.
It is very effective even at level 1, but you may want to use it more as a sidearm or close range weapon while it is low level. It is infamously effective as a sidearm because with the SMG ULM mod at level V, it adds 1 or 2% to the refresh speed (ie it's practically the same as not carrying a sidearm to your refresh speed.) Without the ULM mod (which you may not have at this stage) it is a fairly weighty SMG at level 1, but that's hardly an impediment, given the impressive damage output.
You can offset the barrel climb with the thumbstick, once you get used to it, and maintain decent accuracy that way, but it takes a bit of practice.