If the current project has been poorly managed so far & dates keep slipping, that is likely to continue to be the case. The launch date could slip even further.
You already arranged a leave date with your employer, and you already went beyond what you were required to by trying to schedule that so you could help launch the project. You are not responsible for their failure to hit their schedules.
Trying to push your start date back with your new employer could also damage your upcoming relationship with them: they're more likely to care that the date change hurts their plans then that you want to do something generous for your current employer.
You already arranged a leave date with your employer, and you already went beyond what you were required to by trying to schedule that so you could help launch the project. You are not responsible for their failure to hit their schedules.
Trying to push your start date back with your new employer could also damage your upcoming relationship with them: they're more likely to care that the date change hurts their plans then that you want to do something generous for your current employer.
I think you should get out.