Doesn't look that serious so really just needs filling. What's the planned finish for the guitar? A solid paint colour and I'd use a thick superglue. Any glue is going to get onto the surface so will ruin it for staining. Otherwise I'd use a wood filler, not a glue, just to fill the cracks and get a level surface.

Once you get a finish on it, the wood will stop reacting so much to temperature and humidity changes.

Looks like you've got some sanding to do to get rid of the machining marks, so you may well reduce the crack depth to almost nothing at the end of that process.