Why did the Enterprise drop out of warp before reaching Bersallis III?

In short, warping dentro a solar system is generally considered a last-ditch maneuver. It's certainly possible (and does occur in a number of episodes) but at the very least it seems to be highly inadvisable, presumably due to the large number of potential hazards such as interstellar objects and strong gravitational fields:

DAX : We're too far.

KIRA : Wanna bet? Take us to warp.

DAX : (concerned) Inside a solar system?

KIRA : If we don't, there won't be a solar system left.

_{^{DS9 : By Inferno's Light}}

e

TUCKER: You're going to warp?

PAXTON: A five second burst.

T'POL: (incredulous) Inside the system?

_{^{Enterprise : Demons}}

And even when the fate of the human race is at stake, the Enterprise doesn't risk an in-system warp jump:

WESLEY : Enterprise approaching the Terran system, sir...

RIKER : Slow to impulse... time to intercept [the Borg]...

WESLEY : Twenty-three minutes, fourteen seconds, sir.

_{^{TNG : Best of Both World, Part II}}

Although it's clear that the Enterprise will be cutting it fine getting to Bersallis III, it's evidently not worth risking a thousand lives trying to warp across the system merely to shave some time off of their estimated arrival.

Large objects like Planets and Stars can interfere with Warp travel, and in addition, sustained warp drive causes damage to subspace, as well as space as a whole as detailed in Force of Nature.

"Until we can find a way to counteract the warp field effect, the Council feels our best course is to slow the damage as much as possible. Therefore, areas of space found susceptible to warp fields will be restricted to essential travel only, and effective immediately, all Federation vessels will be limited to a speed of – Warp 5? – except in cases of extreme emergency."

That's Picard reading out Federation mandates that restrict warp travel.

However, the episode you are asking about takes place antes the restrictions are put in place, so this alone can't explain the reason behind their premature drop out of warp.

However, if Starfleet is willing to impose safety limitations like this for the dangers of prolonged warp field usage, they're likely to use similar restrictions where warp failure would be catastrophic - for example, near an inhabited planet.