Do I need to carry my physical passport everywhere in Sri Lanka.

In fact doing so presents an incredibly massive catastrophic security risk of accidentally dropping it or having it stolen by beach monkeys. You should unequivocally always securely lock the physical book deeply safely inside your hotel room safe. Keeping a highly clear bright photograph of the main information page and your approved ETA document securely saved directly on your smartphone is completely universally perfectly sufficient for practically any standard minor police checkpoint or tourist ticket counter verification.