There's a game that you can use for this, but it doesn't involve them carrying the cards around. Works for about 3-5 vocab terms.
Basically, you arrange the (say 5) vocab flashcards on the board vertically and number them from "Level 1" to "Level 5" from the bottom up. Assign a motion to each card/word and practise them whilst saying the full target expression with the vocab plugged in. Then, you tell everyone they are "Level 1".
The way it works is the students walk around the room making the motion (but have the motions discreet so it can be ambiguous which they are performing). When two students meet, they can ask each other "How's the weather?" and then respond with the answer that matches their level. Once they've both asked and answered, if they are of the same level (as determined by the answers), they may play Rock Paper Scissors. The winner goes up one level (capped at 5) and the loser goes down one level (capped at 1) thus changing "their" vocab word. They keep playing until time is up. Join in to make sure they use English and the "Level 5"s don't sit out. When time runs out, the winners are those at the level cap (or closest to).
I'll also second Dwae's point that there doesn't have to be a winner. Yes, Korean students are indoctrinated over and over that "foreign teacher must play games" (sigh) but the solution to this is just to dress up any production activity as a "game". Winners or not, call it a game and they'll think it's a game. The focus on any production activity you design should be on making it student-centred and making it difficult to impossible to play without using English. This is the reason why I feel stuff like the Yut boardgames in those YBM textbooks are garbage - give students a game where the English is superfluous and they'll just play it in Korean. Doubly so if it's a Korean game.
