I haven't done the maths, but IIRC Drag is a squared function (although the square might be velocity, not pressure) - so I'm not sure your tenfold reduction is correct.
Anyways - to answer your question - firstly: No Shotguns would not be more prevalent.
Firstly, we need to consider the premise of the question (this is why I didn't both with the maths) - We assume that both Rifle, Pistol and Shotgun rounds all suffer the same penalty for the increase in pressure (this isn't strictly true, I'll cover that in a second) - So, let's run some numbers:
Max Effective range for Double-Ought Buckshot is about 50-60 yards.
Max Effective range being a combination of accuracy, terminal ballistics etc - it's not the maximum lethal range.
Max Effective range for 5.56 is around 500-600 yards
So, let's assume that everything is reduce in effective range by 10:
Shotguns now are lethal out to 5 Yards, whereas 5.56 is lethal out to the same range as Shotguns.
You'd get more utility out of a Claymore sword than you would a Shotgun.
But wait! There's more! Remember how I said I was being lazy and assuming everything suffers the same penalty? Well, Shotgun pellets are balls. They are less aerodynamic than a Rifle Projectile.
Using our same comparison - a Shotgun Pellet has a drag coefficient (as a sphere) of about 0.4, whereas a 5.56 round has a drag coefficient of about 0.28 - therefore shotgun pellets would suffer more than Rifle rounds.
To give an answer to the second part - how would Firearms develop differently?
My answer is that you would see longer, thinner projectiles with a greater mass than current rounds. This is because it is a more aerodynamic shape. Possible you'd even see sub-calibre munitions, like an APFS-DS rounds from Tanks (Armor piercing, Fin Stabilized, Discarding Sabot) - essentially giant tungsten darts.
This would be so that there is minimal drag and in order to maximize terminal ballistics, you have a greater mass (therefore greater momentum) to offset the higher rate of deceleration once the round leaves the muzzle.
This would significantly increase recoil - so a move to heavier firearms or perhaps a reliance on fixed emplacements, rather than man-portable firearms might be common.
Other options would be an earlier switch to man-portable cannons, where the terminal ballistics isn't reliant on velocity, but is due to it being a cannon shell with a warhead (so it goes boom when it impacts on target).
However - there is another possibility, and I'm raising it because I'm both a WH40K nerd (and the Bolter is cool) but also it would be better in this scenario - which is the Gyrojet family of Firearms.
Now, granted in reality they never took off because it was new, the bugs weren't ironed out and it didn't do anything better than existing technology.
In your context though - having a low-velocity (and therefore low recoil) heavyweight projectile that has it's own method of propulsion (so it maintains it's velocity over a greater distance) would have such a clear advantage over 'conventional' firearms in terms of range that this innovation would have been developed further. Before anyone points out - yes, I know the Gyrojet had an effective range of only 50-60 yards - but that was limited by the amount of fuel in the rocket - more fuel = more range, and the possibility of having a larger projectile that was powered for significant portion of it's flight time to hit out to targets 4-500 metres away would be game changing.
Shotguns, being relatively short range, would be even shorter range to the point that they are essentially a Melee weapon. Rifles would still perform better due to ballistics. We would use different styles of Bullets for better performance and most likely Gyrojet technology would be adopted and used.