Only spells with a Somatic component require that you have a free hand.
If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures. - Player's Handbook, pg. 203
War Caster states:
You can perform the somatic components of spells even when you have weapons or a shield in one or both hands. - Player's Handbook, pg. 170
It does seem a little odd that the War Caster rules would bother talking about having only one hand full. I believe the purpose of the War Caster feat is to allow you to perform somatic components when you have both hands full.