Is resigning from a job—due to being certain that a colleague is stealing and the manager taking no positive stance on the matter—considered severing one's livelihood by one's own hand, especially when there is no alternative employment?

A person is obligated to seek lawful sustenance. If they find that a certain job threatens their future, there is no sin in leaving it. It is recommended to consult those who offer sound advice and to seek guidance from Allah (istikharah) by performing two non-obligatory rak'ahs of prayer and reciting the well-known Istikhara supplication. As for the mentioned saying, we have no knowledge of it, nor is there any evidence for it.