You don't want to run a servo from the same power supply your PIC runs on unless it's a bench type power supply or one that can handle a ton of current without dropping its output.

Use a separate supply or batteries for the servo motor. Should take care of the problem.

You might also want to strap a 0.1uF cap across the PIC Vcc & gnd pins too. Helps filter spikes from the supply, and it can keep the PIC from resetting at odd times.