I think it might be cause of the rapid change of his living conditions, he has moved from 3 different household in the last 3 months. The only person that isn't new to him is me, so could this be part of the reason for the sudden pooping disorder?

In a word - yes. His world has been badly shaken and he needs time to adjust.

Absolutely. It may take a little longer for him to get adjusted to his new surroundings, and you may even need to go back to basics as far as training goes. So as soon as he eats or drinks, he goes out side, a small party when he does his business, maybe a kernel of his kibble for a treat and lots of "good boys"

Have you noticed any other changes in his diet- eating, drinking, exercising? If it continues or anything else with his health changes, consult your vet. Try to make him comfortable, and create things that may be familiar to him, and take him outside.

Maybe something is scaring him out side...

My shih Tzu has been pooping at night on the bedroom rug. He is a
Rescue and is 5 - 7 yrs. Old. I have had him for 2 years and this just
Started. He is well behaved and has a CDC certificate so I can take
Him to Hospice and nursing homes. Nothing has changed in our lives.