Even you can't use overt games, you can still do plenty of them disguised as repetition drills etc.

By giving each row a team name, something as simple as having each student in the row repeat a target sentence can become a (relatively) exciting team-based timed competitions.

Here are some warm-up activities I use that you could pass off as not-a-game that don't required students to physically interact or leave their desks:
 Use timers for teams to make them competitive! I usually have the active team stand, with students sitting when they are done with their contribution.
- Snow ball: give first student a word, they need to make a new one using last letter of your word as the first of their own. 2nd student does the same thing etc.
- Stream of consciousness: show a random ppt photo. First student things of a related word, second student thinks of word related to 1st student's etc. ie apple-->red-->blood-->vampire
- Topics: Student 1 thinks of a topic, each student comes up with an example
- Shopping lists: student 1 says a vocab word. Student 2 repeats it and adds their own, student 3 repeats both and adds their own etc
- etc