Are there any ships where three adults can share a suite?

Yes, all cruise ships have cabins/staterooms and suites that can be shared by three adults, either with a pull-out bed, an extra bed or a sofa bed. However depending on which cruise line you book with, you may find space to be cramped. Often cruise lines will offer large discounts for third or fourth adults sharing a room, so you may find that the financial benefits far outweigh any issues with space.