Does increasing a ship's tonnage necessarily give it a deeper draft?
Short answer
No. Hull shape, displaced volume, and the internal distribution of weight determine draft, so a relatively heavy ship can still be designed with a shallow draft.
Notes
A ship of several thousand tons can still have a shallow draft. The hull's shape controls how it displaces water. The arrangement of weight inside the hull also affects how it sits. Tonnage alone leaves out these design choices.