I’ve been trying to figure it out but I just get more confused. Can anyone help me understand?

Yes, why wouldn't it?

I was under the impression that if both hands were full, you wouldn’t have a free hand for the somatic aspect of the spell

When you cast a spell with a Material component, the hand holding it can also be used for the Somatic component.

It works because you can use the same hand for the material component (which is your weapon) to also do the somatic component

Since it has both somatic and material components you can do both with the weapon bearing hand.

Shield spell is the difficult one without War Caster.

So follow up question, the war caster feat says “Somatic Components. You can perform the Somatic components of spells even when you have weapons or a Shield in one or both hands.” Is that ONLY for spells that don’t have a material component, but DO have a somatic component?

Yes, it would be redundant for any spell with a Material component.

I would say it would work with free action-drop weapon, but RAW you can't use a free action outside of your turn. Some DMs may grant you that but it's homebrew territory at this point. Dropping a weapon ON your turn in anticipation of potentially needing Shield can work, but an enemy can pick it up, kick it... I guess a mace on a wrist leather strap might work tho. You just won't be entitled to opportunity attacks unless you just punch/kick, but if you were weapon-dropping for Shield it's unlikely you'll switch to opportunity attack at the last second.

Side note, you only need two hands to actually make an attack with a two-handed weapon. You can hold it with just one hand, e.g. while casting a spell.

🤦🏾‍♂️

OMG

Not just any material component, specifically material components that are your weapon or shield

Cry about it I guess