There is no general relativity involved in calculating your position. Solving for your position is basic geometry, if you want to have better resolution, have good model for the effects of the atmosphere on the radio waves.
In "Exploring Black Holes" by Taylor & Wheeler, project A is titled 'Global Positioning System', where they say locating your position accurately depends crucially on General Relativity.
Before the satellites are launched, their atomic clocks have to be tuned to offset for time dilation.
GPS satellites are all tuned this way, and it has the desired effect.
However, for people implementing GPS receivers, the signal arrives already corrected for time dilation. So the people making that part of the system don't have to know about general relativity at all :)
Without correction the drift due to combined coarse relativistic effects would be on the order of 100,000 nanoseconds per DAY.
In fact the first NTS-2 satellite was launched with the clock unadjusted for GR, due to doubters, but after three weeks the syntheziser had to be activated to compensate. The calculated uncorrected drift on that clock alone was 38,000 nanoseconds per day.
As the document notes, to further refine Navstar to greater levels of precision would require tackling yet another set of tougher relativistic effects, plus more mundane atmospheric influences.
