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.
No, you can't cast it using a spell slot.
Spells you can cast because of your race aren't spells you know, and thus can't be spells you prepare. Per the PHB (p. 201, "Known and Prepared Spells"):
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind ...
... and it goes on to describe some of the class-specific variations of knowing and preparing spells. But in any case, having a feature from your race or class that allows you to cast a spell isn't the same as knowing the spell.
This is spelled out clearly in the multiclassing rules for Pact Magic quoted in the question (PHB p. 164; emphasis mine).
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
Likewise, the Drow Magic trait clearly distinguishes the known cantrip from the merely castable other spells (PHB, p. 24; bold for emphasis mine):
You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.
If the intent of the rules was that Drow also knew the non-cantrip spells, it would say so.
Best Answer
The only way, RAW, to get a feat at first level, is to be a Variant Human. PHB page 31, "Variant Human Traits"
If your DM is open to houserules, however, they could simply give all characters a feat if they choose.