Can molluscum contagiosum be transmitted by touching the lesion and then touching others, if the lesion is completely intact and not been scratched?

Molluscum. Molluscum contagiosum is transmitted by direct contact, either person to person or by shared items, such as clothing, towels, and washcloths. Mc can be spread by contact sports. Furthermore, if a person touches the rash and then another part of his or her body, it can cause spread to that area (known as autoinoculation).
Yes. As long as there are lesions on the body it can be spread.