You're looking at it slightly wrong : voltage affects speed, but it's closer to say that torque affects current than vice-versa.
So run the motor unloaded : at 12V it'll run at about 2/3 the speed at 18V, that being the speed at which it generates enough back EMF to cancel most of the driving voltage. (Datasheet says 15200 vs 24000 at 18V).
The rest of the driving voltage (maybe 10 or 20%) is dropped across the motor's winding resistance, resulting in two effects : enough torque to turn the unloaded motor that fast, and heating the motor, by I^2R watts. You can't measure that voltage directly, but you can measure the resistance and current, and I recommend you do. I'm guessing about 1 ohm, and the spec gives a no-load current of 400mA, so 0.16W in the windings (and 4.8W total, so the rest is lost in friction in brushes and windage).
Now add a little load. As you add a torque load, the motor will slow, so the back EMF will reduce, and the current will increase until it supplies the torque you need. This is why I say torque affects current... Efficiency also drops too, because that extra current creates more heat. You don't want to use the motor below 75% of its unloaded speed - maybe 60% for short periods then cool it off.
Now add your foam wheel. You haven't told us but I'm guessing it's quite large, and simply spinning, you're not running it across the floor. How fast does it run? You haven't told us, but I'm guessing its wind resistance is acting as a huge airbrake, so a few hundred RPM.
Which is pretty much stalling the motor. It can't generate any back EMF, and virtually all of that 12V is developed across the winding resistance. So I=12A, power = 144W wasted as heat. Fortunately your PSU isn't up to that, so you may still have a working motor...
You need a high torque, low speed motor. Or, more likely, gearing. This will increase the motor speed and reduce the torque demand to let the motor drive the wheel efficiently.