The defendant, dissatisfied with the verdict, appealed without hesitation.

Literal

Defendant [topic-は] verdict [to-に] dissatisfied [so-で] without-hesitation appealed.

不服で means 'being dissatisfied' — the で marks the reason. ためらわずに means 'without hesitation,' using ~ずに (literary negative adverbial, equivalent to ~ないで). 控訴する means 'to appeal (a court decision).'