I also face this often. Thing is, there isn't only high food - there's either lots of food, spread across lots of tiles (unirrigated rice, flood plains, sheep, ...) and big food surplus cities (irrigated corn, grass pigs, ...).
Usually if I have a city with lots of FPs I cottage it up and leave the palace where it is. It depends on the map though, I rely on what early scouting yields; if there's no high food surplus site I consider moving the palace and use my old capital as a GP site.
If I have a capital with 3-4 high food surplus tiles, I generally move my palace to a FP-site or another nice commerce site and use the old capital as a GPFarm. These are generally very good and as they start with GP generation very early, they usually give me quite a lead GS wise.