Where can I double check my elliptic curve results?
The slope is $5$:
$$ \frac{27}{4} = 27 \cdot 4^{-1} = 27 \cdot 2 = 5 \mod 7 $$

$$ x = 5 \cdot 5 - 3 - 3 = 19 = 5 \mod 7 $$

$$ y = 5 \cdot (3 - 5) - 2 = 5 \cdot (-2) - 2 =  5 \cdot 5 + 5 = 30 = 2 \mod 7$$

So $(3,2) + (3,2) = (5,2)$. This matches with $(3,2) + (5,5) = (3,5)$, since this shows that $(3,2)$ and $(5,5)$ are on a line which is tangent to the curve at $(3,2)$.