Are there any true non toxic wood glues?

Probably every carpenter in North America uses, and always has used, Elmer's Carpentry or Titebond.  Both are non-toxic.  As for strength, that's what they use in laminate beams.. Not Elmer's White - Elmer's Carpentry.

Aliphatic glue - White glue, Elmer's- is completely non toxic. There are people who eat it, for some reason. Sort of a fetish. It's the go-to glue for many wood workers, too.

You can use rabbit skin glue or genuine hide glues - and for fine work it is recommended as it allows future repairs to be made. The real off-gassing comes from the glues used to manufacture sheet materials, and you should avoid those first. PVAs will generally release some formaldehyde, but it is only significant if you are sensitive. 

Just regular white glue, like elmers is not toxic.

Unless you EAT them, most glues are not toxic.