Is there any significance to starting with the left side when putting on clothes or shoes, or when leaving the house?

What you are experiencing indicates a lack of attention, habituation, and care for the Sunnah of prioritizing the right side in showing honor, which is a Sunnah based on what the two Sheikhs (Bukhari and Muslim) reported from Aisha, may Allah be pleased with her, that the Prophet, peace and blessings be upon him, loved to use his right side in all his affairs: in putting on his sandals, combing his hair, and purifying himself.