I get a lot of calls from strangers. They only want to sell me things, not change my family dynamic, but even from that point of view, I can tell you that your current plan is unlikely to be the smoothest possible approach. I can suggest some improvements.
First, don't ask "do you have time to talk?" That is not a yes or no question. It depends what you want to talk about. Selling me life insurance? Changing my mortgage provider? My Google Business Listing? No it's not a good time and it never will be. And those are the people who usually start with that question. Just skip that and go straight on.
Second, asking if they have a son named ABC such and such an age - this sort of thing strikes terror into a parent's heart. Are you calling me from a hospital or a police station? What's going on? Don't start there. Plus, you're taking away their ability to disengage from you once they know what's happening. "But YOU SAID you had a son that age with that name! You must be the person I want!"
Just start where you need to start.
My name is EFG. I'm looking for ABC. I believe he is my father.
(Note: not "I am his daughter." This call is about ABC, not you.) And no matter how sure you are, don't state it as a bald fact.
Third, don't end the call when you've got your confirmation and switch to a letter. More on that in a moment.
At this point, right after you've said "I believe he is my father" a lot of stuff could happen. The 70-something person could just hang up on you. They could start asking you questions. They could tell you that ABC died years ago or is estranged from them or otherwise unreachable - and that might or might not be true. It's pretty hard to plan the conversation past this point, but you should be prepared to answer questions if they are asked. Are you looking for money? Are you looking for an organ donor? Do you want a medical history? Do you want to meet anyone other than ABC? If ABC is not reachable, would you like pictures of him or other things the family can provide? Are you angry at ABC or his family? How is your mother doing? Does she want to meet anyone? Do you want to be friends on Facebook with this person you're talking to now, or with ABC? Are you prepared to offer your phone number, email address, social media handles and so on to enable continued contact? Think about the things you might be asked now that you've started this conversation. [Note: I'm not asking you any of that. But the person on the phone is likely to and you need to know your answers before you start this.]
If things get emotional at all, apologize. "I'm sorry for disturbing you. I really want to be in touch but I understand a phone call out of the blue can be difficult. Can I call again in a few days?" Then listen. Abruptly ending the call when you've got what you wanted and carrying on in a letter is not very sensitive. If the person wants to end the call, end the call, and call back if you got permission to. If they're ok with still talking, you can still talk. If they want to switch to another medium, or pass you along to ABC and not talk to you again, respect that.
I have a little experience with this in the form of a cousin I met when he was in his teens. At that point his father had only known him about 5 years. My family was generally cool with it, it was not a huge disruption or a big deal. ABCs family may be thrilled to gain another person or may be embarrassed, ashamed, or angry and suspicious. Nobody can predict that. So knowing what you want and why you are seeking contact will help you to stay grounded if things get emotional (for good or bad) quickly and also to answer a possible barrage of questions in a way that doesn't make people think you're up to no good.
