...I read a lot of accounts that say how scarce lead was in many cases...
As mentioned, shot was generally available. If lead was scarce in certain situations, snares and traps would likely be used on small animals and lead conserved for bigger game or protection. Right tool for the job.
To put some numbers to it, from T. M. Hamilton's
Colonial Frontier Guns conscerning French shipments to Lousiana:
Hamilton said:
...in 1733, the annual shipment to Lousiana included 20,000 livres of bullets weighing 28 to 32 to the livre, 10,000 livres of shot and 20,000 gunflints...
That's 10.8 tons (US) of balls and 5.4 tons of shot.
Hamilton said:
The following year, in 1734, the French shipped in 20,000 livres of balls weighing from 25 to 28 to the livre, 30,000 livres weighing from 28 to 32 to the livre, and 28,000 livres of shot "for wild geese, ducks, doves, royal and demi-royal" and 100,000 gun flints...
That's 27.0 tons of balls and 15.1 tons of shot.
These shipments were for Southern Indians, but their use of shot in the period appears to be pretty much the same as whites. That is "for wild geese, ducks, doves" or in other words--fowl.
I'm sure there are lead ore deposits back east, but I'm not familiar with them. Missouri on the other has significant lead ore deposits, and the French first started mining and smelting operations there in 1720. Shot towers were built as early as 1813. The first major American urban shot tower was erected in Philadelphia in 1808.
As
tenngun said, swan shot was a size of shot not the shape.
Again, referring to Hamilton's book, the first shot made was done by cutting sheet lead into cubes, then tumbling the cubes in a barrel to round the corners.
In 1665, a new technique called "Rupert shot" for Prince Rupert was developed by dropping melted lead through a colander-like vessel a short distance distance into water to solidify it. The shot made with this technique was not perfectly round, but slightly tear-drop shaped.
William Watts, of Bristol, invented the technique of making almost perfectly round shot by dropping lead through a colander or screens from a great height before hitting water to cool and solidify them. The extra distance the molten drops of lead fell allowed the surface tension of the lead to shape it into spheres. The dates I found for Watts invention varied from 1769 (earliest date the idea came to him in a dream) to 1782 (the date he received a patent for the technique).