The one thing you will notice is that you will probably have a faster idle with the big block. So taking off from a stop will be somewhat different, but with a 5 speed and the 3.73 rear end, it will be just a minor difference.
Also, the steering effort at slow speeds will increase due to the increased weight.
The one place where the drive-ability will change dramatically, will be at WOT
You will be able to make the Cobra lose traction and spin out much easier with the torque monster.
But go for the big block. Enjoy! You did not buy the Cobra for good gas mileage! Take the jump. If and when I blow up my 425HP small block, I will go the big block for a replacement.
I have never heard anyone on this forum say that he (or she) had too much horsepower!