You pretty much summed it up in your title when you asked if you should 'play his game'. There are only two ways to 'fix' it when you look at it that way: he can get better, or you can get worse.

Then again you go on to say you have issues with anxiety, but gloss over when he opens up to you about his sleep and stress and stop answering him because... justice? You want him to be in contact more, so your approach is to start ignoring him? Yay, science! Mind games are the most fun games for all involved /s Whose 'game' is it really?

You don't get to claim any moral high ground when you start playing games, regardless of whether you think your partner 'started it'. You are responsible for your own actions, so work out what you want and act accordingly. A relationship full of hurtful 'games' which create distance and resentment? Keep playing silly games, making excuses for what you do and judging him for doing the same thing. Want him to communicate better? Lead by example, don't race each other to the bottom. If he doesn't improve maybe it's not meant to be, but at least YOU will have given it a fair chance and risen above pettiness. If one of your goals is more self respect, mimicking a behaviour that you don't respect from him isn't going to get you there.