I don't play elizabeth, but you have the same problem with K'.
See, when you usually do a dp on a stick, you'll sometimes hit forward. Like this
. When you try to do qcf
, the game registers that as a shortcut because the qcf in the previous motion is read. So it does the super.
What you can do are three options:
You can do
and double tap the button very fast. This requires a bit of timing on button pressing, some have even suggested using the same strategy SSF4 players use for plinking, but how you do it is up to you.
You can shortcut the DP by making it hcb,f+P. This is not really a shortcut, but a LONGcut. For Elizabeth, you can do hcb, uf+P or hcb,df+P and that should give you the dp without doing her command grab.
The last method is perfect execution of the move. That means, without letting go of the stick, you go do the DP into the fireball motion. That takes some practice as you have to move the stick backwards to neutral down and do your fireball, but it's possible as long as you don't let the stick go (It usually hits forward if you do.)
It all requires practice, but I'm sure you're no stranger to it. Hope this helps!