Can tight scalene muscles cause a prickly/sore sensation across shoulders and upper back?

Neck. The scalenes muscles in and of themselves may not cause this sensation. The other muscles or the upper back and shoulders such as trapezius, splenius capitus, and accessory muscles of the shoulder that support the scapula may cause this feeling. If doing rehab for neck tightness all of this areas should be addressed as well as posture and sleeping habits.
Yes. It sure can. Try massage, a heating pad, or stretching/yoga to help with this. If it's persistent, you may need a muscle relaxer and/or physical therapy.