it doesn’t have the waiver exempt badge on his name, but he’s also never played in an NHL game before so I would think he would be exempt

Nope, he's waiver eligible, waiver eligibility is measured in games played or seasons, Gawanke may have never played an NHL game but he's spent enough seasons under NHL contract to be waiver eligible

Edit: In fact if you look at his history, he was placed on waivers last year by Winnipeg