Now you made me read up on the genetics.

Lefties and ambidextrous are appear to be genetically mostly independent. With lefties being a much more complex inheritance with multiple genotypes.

https://sci-hub.st/10.1038/s41562-020-00956-y

Lefties also tend to have a higher rate of bipolar and schizophrenia so about that not being right in the head... They also tend to have an gene for increase in inter-cranial volume.

Standard disclaimer: With human genetics the results always have to be taken with a grain of salt. A few annoying things like ethics and time prevents cleanly associating cause/effect. So human genetics use mostly correlation studies with their inherent inaccuracy.