My core question is what can I do or do I have to do to ensure the
success of the project after my departure? Further then mitigate the
risk of the bus factor.
That is not your issue. It is the issue of the management, they have to ensure that the bus factor is on adequate level, not you. If bad management caused almost the whole team to leave, than it is not your fault that the bus factor becomes 1.
I came to the conclusion that it's better if I don't say anything
until I've signed the new contract?
You could at least talk to your seniors/your management and say that you are very unhappy with the described situation (don't mention to leave yet). I predict that instead of a real attempt to enhance the situation in the long run you get either lip-service promises that they will change the situation (which they won't) or they will be upset. Both will verify your intention to leave but if you do it this way no one could claim that you did not show any signs for leaving (which is also not an obligation, rather a manner).
If you have the confirmation of your new employer that you are accepted (that is before even you get the new contract) you submit your notice.
Could it be seen in a bad light by future employers when I leave the
"sinking ship" or could it even be seen as if I brought the ship to
sink?
Let me ask this question: Do future employers need to know why you left this company? And even if they want, tell'em the truth: Management messed up and could not motivate and preserve a team for project and is unable to plan in the long run. Instead of hiring new staff, you get more and more tasks of the resigned team which affect your health.
An employer who does not understand this as a feasible reason for your leave of the previous company is not worthy to enjoy your presence.
I came to the conclusion that it's better if I don't say anything
until I've singed the new contract?
Again, that is not your obligation, it is the managements task to secure an adequate bus factor in the long run.
CONCLUSION:
Companies emerge, (bad) companies go and that often due to reasons on which you do not have any influence (like in your case, at least it seems to me).
There is no reason why you should invest any time or your nerves for attempts to save what cannot be saved. Besides, management and HR departments calculate your salary strictly "according to the market standards." It is then only justified to let them feel the "market" (better offer, more cash = good bye).