[RPG] Does a wizard need to hold a component pouch or focus in one hand in order for it to work

You're right, you're kind of stuck.

Thankfully, the action economy of 5e works in your favor here.

Here's what you need to do:

• Select a crystal as your Arcane focus
• Wear the crystal on your body (necklace, pendant, whatever). Components pouch also works here as there is no action economy on drawing stowing one of those.
• When you want to cast a spell, drop your weapon and touch your crystal
• When you're done casting, pick up your weapon.

This takes no additional actions, is perfectly legal (nothing specifies how big a crystal must be, just that you have a hand free to use it. (dropping a weapon costs you nothing, picking it up is your interact with object that's free with an attack or move).

The only real risk here is that if you do this a great deal, an enemy might begin to ready actions to attack you, or even move away from you while your weapon is stowed. This is something you'll have to live with for this to actually work.

Last thing that I'll say is that you should talk to your Adv League GM, or even seek out an answer from their twitter feed on this. GMs for Adv League are supposed to be lenient and use common sense in their rulings as the ruleset is flexible by design.

You can't cast spells while holding your two-handed sword or your sword and shield without the War Caster feat. This is spelled out on page 203 of the PHB:

If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.

However, you don't necessarily need to take the feat in order to use a two-handed sword or a sword and shield and cast spells. (I realise the previous sentence seems to contradict the first one, but I'm getting there.)

Each turn, you can interact with one object without using an action. (See page 190 of the PHB for details.) If you are using a sword and shield, this means you can, for example, sheathe your sword without using an action, then cast a spell on the same turn. Then next turn, when you want to attack with your sword, you can draw it without using an action, then attack on the same turn. Note that this leaves you unable to take opportunity attacks between these 2 turns.

You can find more information about wielding weapons and casting spells without War Caster here, here, and here.