How to split text between separator into multiple files?

I have a text file contaning the following:

"random
textA"
"random
random
textB"

The separator is "

How can I split the containt into multiple file as follow using a bash command?

File 1 :

random
textA

File 2 :

random
random
textB

I came into examples using csplit or awk but they does not cover this text layout.