Indeed, it seems the saucer stays at rather high warp for at least the better part of a day in "Encounter at Farpoint", even though our heroes think the saucer would slow to sublight in just two minutes if forced to separate in "Brothers".
Of course, in the first case, our heroes wanted the saucer to stay at warp; in the second, they wanted it to drop to sublight. For all we know, the saucer can stay at warp indefinitely if need be, by maintaining a warp field with some means other than actual warp engines. Or alternatively the saucer has actual warp engines - this has never been denied in dialogue or plotlines, after all.
In "Arsenal of Freedom", the saucer is sent on an interstellar journey without using the stardive section for giving it an initial warp boost. This might suggest that the saucer can accelerate to warp all on its own; if it couldn't, it would have been criminally incompetent of LaForge in "Arsenal of Freedom" not to give the saucer the sort of boost that allowed a warp-speed interstellar journey in "Encounter at Farpoint".
On the basic issue of clean "warp separation", supposedly torpedoes launched at warp do it all the time. They might have their own warp field sustaining systems, considering that they also move at warp from launching ship to target sometimes. But we know that lifepods can do a warp separation as well - we even see one in "Bounty" and IIRC in "First Flight" as well. It sounds a bit unlikely that such things would have full warp engines aboard; probably some very simple gadget allows an object to leave a warp field, and only those objects that lack this simple and small gadget suffer the fate of the cable that snapped when Tucker moved from starship to starship at warp five in "Divergence"...
Timo Saloniemi