I like most of Cabijista’s version but have a couple of tiny suggestions:

Conflicts erupt in the early days of the Clone Wars.

Having successfully protected Duchess Satine from Death Watch, Anakin and Obi-Wan now rejoin the clone fleet in their pursuit of General Grievous.
(successfully safeguarded > successfully protected, to reduce the number of “s” words in close proximity)
(cut “main” because it just seems a little wordy and they weren’t seemingly with the fleet at all previously)

Meanwhile, the Republic has learned that the assault plan against the Malevolence was leaked to Count Dooku, raising fears of a Separatist spy.
(“have” > “has” learned and “their” > “the” - because “the Republic” is usually a singular noun)
(“arising” > “raising” - grammatical fix)

The covetous NUTE GUNRAY has been caught sending Ryloth’s military secrets to his fellow Separatists, and is being escorted to jail.
(Not sure about the name-drop of Ryloth, since it’s not been mentioned yet, but I understand that you are setting up the Ryloth battle to come)

Master LUMINARA UNDULI has asked Ahsoka to help question Gunray, as they hope to uncover both Grievous’ location and Dooku’s plans.
(removed the comma before “and Dooku’s plans”)

But Dooku will not allow that information to fall into enemy hands…