I used the same shortcode to inject the same CF7 form twice on my webpage.
I managed to kill this issue creating a second form and using two différent shortcodes.
I suspect jquery to get lost with two forms using the same ID.
Try this if you use a the same form twice on a webpage :
1. Clone your form
2. Get the shortcode for this clone
3. Replace the 2nd occurrence by the clone’s one.
It May do the job ??