Wow, with 12 hands at once? How many chances does each get to answer and draw again?
TBF, it wasn't done by draw poker, but by drafting.
Imagine a set of cards available in a draft row. Starting with the best answer a team pulls a card from the row and choose to either add it to their hand or to burn it for the top of the deck. (This is to allow meaningful draft choices even when none of the cards directly help you) There are only a limited number of drafts though, so the teams at the bottom of the order only get to top-deck blind.
It's very easy to set it up for later rounds to be worth more by doing things like adding jokers to the game or by increasing the number of cards available to teams. A typical round might be: each team gets dealt two cards blind, then there are three rounds of questions with community cards revealed in between question, so that each teams has seven cards at the end, but with teams who do better at the trivia having more control.
Because of the order in which rule variants are structured from round to round, a winning hand in the first round might be two pair, but a winning hand in the third and final round might be a full house or four of a kind. So that provides a natural ramp-up by simply making it easier to make bigger hands in later rounds.
I have a very large deck (something like 11x14) of cards for props and drawing, but once the cards are taken, they're represented in an on-screen display, so everyone can see everyone's hand at all times. It's all a pretty low-tech thing that I did over ten years ago. I've got a spreadsheet attached to a PowerPoint using the
PowerShow add-on. Normally when you embed a spreadsheet in a powerpoint presentation, the presentation doesn't update live when you update the spreadsheet. The add-on allows that to happen. So I have two tabs in a spreadsheet, one where I can enter something like "TH" in a space and another where a Ten of Hearts appears in a card deck font. And that second screen is embedded in the PowerPoint. So I type the rank and suit in a space and a card appears on the screen in the matching place for the right team.