Yes, if you discard a 1 carry bit then that signals overflow. Don't worry about overflow. It is correct to discard the carry bit.
But since when we store the 8-bit representation of these numbers, we lose the MSB, that would flip the signs.
It's important not to think of discarding a 1 bit as flipping the sign. First, discarding a carry bit is not the same as flipping the sign bit. You can discard the carry bit from a negative result and still end up with a negative answer. For instance:
1111111
11111111 (-1)
+ 11111111 (-1)
--------
1 | 11111110 (-2)
The final 1 carry bit is discarded, but the answer's sign doesn't flip.
Second, even if you're thinking of flipping (as opposed to discarding) the sign bit, it's not good to think of that as flipping the sign. In sign magnitude representation flipping the sign bit will flip the sign of the number. But in one's and two's complement, negation is more than just flipping the leftmost bit. If you just flip the bit you get a very different number. Yes, it has the opposite sign, but it's not the same number.
Sign Mag. | One's Compl. | Two's Compl.
01111111 = 127 | 127 | 127
11111111 = -127 | -0 | -1