There's a time element with all of this. If my biological father is that ill he might be the only chance to get the name of my mother. So I can't afford to be as patient as you all want me to be.
Time frame
It's only been three weeks since you got your first email from someone regarding this whole thing. November 5th you heard from them, and it's only November 26th.
Risk analysis
What you're saying is that you're balancing your aggressiveness against the likelihood of a line of inquiry being permanently shut. But if he's on death's door such that a week or two might make the difference, then it's already too late and it's nobody's fault - certainly not your cousin's for taking his time about the whole matter.
What you're essentially betting is that he's sick enough that he's going to die within weeks, but he's well enough that he can help you. And the small chance of this being exactly the case is causing you to close doors and possibly burn bridges just in case he's in that exact situation and no one else has any possible way for you to find anything else out.
The risk of that being the case is low. The cost you're spending to prevent that possible outcome is very high.
What's more important - information or relationships?
I suppose, then, that I am making a bad assumption about your intentions. Your actions say, "I don't really care who I upset in order to get the information I'm looking for - I'm interested in information, not relationships, and if they get upset it's really their own fault for not helping me faster." While I'm sitting here thinking, "It's probably too late to develop any meaningful relationship with dad, so if you want to know him you should probably focus on building relationships with those around him."
Relationships
Reading back through the thread and the email you sent him most recently, it sounds like you're threatening an end run. You're essentially saying, "Hey, I suspect you aren't interested in helping me, and if so well I don't need you and I'll just go through other channels." And you're sending this message, ironically, because you feel like he's not respecting you.
Now it could be years down the line you'll all be at a reunion, everything will be smoothed over, and you'll all laugh about it, or it could be that you'll be no closer to knowing your biological family because in your haste to get to the one person you think might have the answer you've upset them and years later they still refuse contact.
Information
Of course, if you aren't interested in developing relationships with your family, and are really only interested in finding the names and medical histories of your biological parents, then I suppose it doesn't matter and you should go full steam ahead. I have a hard time understanding how this will help you though, because even if you do get some information - say confirmation that he's your father, and a medical history - is that really enough? You'll still be left with many, many questions.
Conclusion
But this is me viewing something very important to you through a small window. What I'm seeing - going from "Hey, I might have just met a cousin!" to "Hey, I might have just offended a cousin!" in three weeks - suggests that your current plan isn't going to provide you with long term success, which may depend largely on whether living relatives will want to communicate with you after your most significant possible connection is gone.
So....
THERAPY[DOUBLEPOST=1480197624,1480197443][/DOUBLEPOST]To be particularly explicit about the threat:
If this is due to a family request I completely understand
Then you say
I do have other avenues of inquiry that I can follow, but will wait a few days before doing this.
So you're essentially saying, "Perhaps people don't want to tell me, but I'll continue pursuing it until I find out."