[RPG] Can a wizard using an arcane focus quarterstaff and a shield still cast somatic component spells

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.

Light shields prevent you from using that hand for spellcasting.

Each entry there tells you what you can do with that hand when a shield is equipped on that arm/hand:

Buckler: Anything, including casting spells (but with some consequences).
Light shield: Hold items.
Heavy shield: Nothing at all.

Light shields let you hold items in that hand that aren't weapons, and that's it. Spellcasting doesn't qualify.