Exactly. The background is transparent (as the Flash animation shows), but the pixels around the letters which are not transparent but light shades of the letters' colors cause the problem you see.
The solution, yeah, do 25 PNGs, or create the animation in Flash instead of creating a GIF animation to import it into an animation program
. Another solution would be to use dark gray or black as the transparent layer, the result will be better but not perfect.