Seems like weak clutching in the limited slip diff? They need to be rebuilt from time to time. Depending on the year and how hard the car has been launched in that amount of time. These are not true "posi" rear ends. A true Posi is the next best thing to a spool. The Stock limited slip diffs from 96ish up the thw 99 switch were not rated as heavy as the latter cars are. That 96 had a stock hp of 205 (I think I remember that?) So source a newer more heavy duty unit, or rebuild that one probably the best suggestions.
In fact as I am typing this, i recall my buddy's 2006 Cobra unlocking a little at the end of a burnout, sometimes on a launch, and definitely at part throttle. We stuck an eaton in there and it has been great for 4 years of nitrous oxide injected, 650+ ft lb fun lol.