you can handle all other notes - that don't fit to the 4 chords you want in your progression - as nonchord notes, as suspensions or nineths or sevenths.
You can listen to music such as ostinatos, passacaglias etc. with always the same bass line. (e.g. Last Christmas WHAM)
How can I write 4 chord progressions without feeling like a hack?
It's not the music that's wrong, it's your ears and your inner imagination of sounds that isn't habituated to the "strange" chords.
Listen to similar songs like Tim is suggesting. You can also listen to Bela Bartok's mikrokosmos or better: play it yourself.
P.S. Sometimes our memory is fixed on tunes or songs based on progressions with secondary domanantsdominants or chains of falling fifths: If you are caught in this prison you better change your concept of 4 chords or try a break out by using a playback of your progression and improvising (singing or whistling) variations to this accompaniment.