How to properly add ECDSA private keys?
Well, the normal rules apply, i.e. $(aG + bG) = (a + b)G$, so as long as you add $a$ and $b$ correctly, everything should work fine. Note that you don't actually have to reduce the result of the addition for the point multiplication to give the same result, however your implementation may require the number to be smaller than the order.

Also make sure that you are working mod the order of the curve, and not mod some other number (e.g. the order of the underlying finite field).