Equal power crossfade
if the two audio signals are totally uncorrelated, the the squares of the two crossfade gains should add to 1.  so, to prevent that dip, your crossfade function will look more like $\sqrt{\frac{1}{2}(1+t)}$ and $\sqrt{\frac{1}{2}(1-t)}$ rather than $\frac{1}{2}(1+t)$ and $\frac{1}{2}(1-t)$  for $-1 \le t \le +1$.  note, for constant power crossfades, at the midpoint of the crossfade ($t=0$), both crossfade gains are at $\sqrt{\frac{1}{2}} \approx 0.707$ rather than at $\frac{1}{2}$.  that's what addresses the dip you want to avoid.

if your two audio signals are somewhere between totally uncorrelated and totally correlated, you need to determine the degree of correlation (that's what cross-correlation is all about), and then the crossfade functions can be determinied analytically.  i have posted to music-dsp a little theory about it.  it's not perfectly rigorous, but i think it works.