Do I need different skis for slalom and giant slalom?

Yes, absolutely. SL and GS skis are fundamentally different tools. SL skis are short (155-165cm) with tight radii (10-13m) for quick, snappy turns between closely-spaced gates. GS skis are long (183-195cm+) with much longer radii (23-30m+) for powerful, arcing turns at higher speeds. A SL ski is unstable at GS speeds and cannot make GS-radius turns. A GS ski is too long and slow-edge-to-edge for SL gate spacing. If you race both disciplines, you need both pairs of skis.