Shower head change caused a problem
If you have the old shower head, consider to re-install it temporarily for the purpose of this exercise.

Run the water through the shower head into a five gallon bucket. Measure the amount of time it takes to reach a mark on the bucket.

Replace the shower head with the hand-held unit. Perform the above test again. If the amount of time for the hand-held is shorter than the fixed head, your water flow through the plumbing to the shower has increased. If the time is greater, you have a lower rate of water flow through the new hand-held shower.

It's not unusual for shower temperatures to vary when a hot water tap is opened elsewhere, if pressure and flow are insufficient and/or the plumbing is configured to allow one pipe to take more of the hot water.

I suspect that your flow rate has changed sufficiently to reduce the hot water flow to that leg of the plumbing. Today's shower heads, both fixed and hand-held are supposed to meet US flow rate requirements:


  Did you know that standard showerheads use 2.5 gallons of water per
  minute (gpm)? Water–saving showerheads that earn the WaterSense label
  must demonstrate that they use no more than 2.0 gpm. The WaterSense
  label also ensures that these products provide a satisfactory shower
  that is equal to or better than conventional showerheads on the
  market.