Tessarael
Adventurer
You can, if the spell has both S,M components, as you can sign the somatic components with material component or spell casting focus in hand. Hence, if you can use a weapon as a spell casting focus (e.g., Bladesinger, Warlock Pact of the Blade, with a Ruby of the War Mage, or using a Quarterstaff as a Druid/Wizard/etc.), or if you can use your shield as a spell casting focus (e.g., Cleric or Druid with holy symbol emblazoned on the shield), then it serves both as the material focus and you can use that same wielding hand to do the somatic component too.Honestly the ability to use your weapon as a material component isn't that necessary. The fact that you can't do your somatic components of your spells without warcaster is the bigger problem.
War Caster is only needed for spells with only somatic S components, and War Caster doesn't get around needing a hand free for material M component spells, if you cannot use a weapon or shield as your spell casting focus.