Hi Tiger Tiger,
As an aside, I know nothing about how the Thames barges plied their trade. The barges here in China may have little draft when empty, but are loaded up until they have little if any freeboot at all. I have seen some that appear to be up/down to the scuppers and are maybe relying on the hatch coamings to keep them afloat. I would imagine that in the pre-regulated days of river haulage, similar practices were common on the Thames.
In answer to your query above, the answer is the same as china, in the London river the PLA allowed minimum freeboard was 3 inch's and about 6 inch's on the coast before the Board of Trade took an interest in their freeboards. The more you carried the more you was paid as the crew were paid 50% of the freight rate per ton after agreed stoppages such as dock dues etc. Have seen loaded Thames Lighters with water on deck. As to leaning over to much in a breeze, well yes as a barge it is dangerous to get water on deck, as your stability curve rapidiliy disappears. So the master would reduce sail area as appropiate, but in relation to thier hull size the working sail areas of trading barges was not that particulary big in comparasion to say schooners and fishing boats. Several barges in trade did capsize , one famousily after a Thame race in a private match, in which both master and owner drowned. One caught its leeboard in shallow water, the leeboard did not break and the barge tipped over on it.