You probably won't avoid the fluctuating entirely, but the airline rep apparently will be more willing to cut you a good deal over the phone, versus the robot on Kayak or Priceline. Insofar as you can probably get the true value of the ticket over the phone rather than waiting out the automated prices on these sites. Its kind of strange if you ask me, but this is more of a new trend to get people to shy away from deal sites.
None of this is accurate. Airlines reps have the same information as you get on the website, and airlines usually charge an extra fee for booking via phone.