There's quite a few that Jon could sing that would be easy on his vocal cords, songs like:

I Will Drive You Home
The Fighter
Real Love
Labor of Love
Diamond Ring
Ain't Running Anymore
Fingerprints

Most of these songs could make for a better acoustic set, however even though it may seem easier for Jon, his voice is still technically left bare and exposed, Jon honestly needs to fully re-train his vocal cords and go through an extensive vocal re-training regime but I don't think Jon can dedicate himself enough or want it enough.

Jon's initial vocal deterioration and decline is down to Jon not taking care of his voice enough, he only ever seeks help when it's too late which was the case back in 2013. His voice wouldn't be this kind of shape if he actually was taking proper care of his voice.