contradictory information, she instead wants to insist on written communication only to create a paper trail.
Your cousin's concerns are perfectly sound. Although telephone is a lot more convenient than writing, a good way to minimise accountability for one's poor conduct is to avoid creating a paper trail to begin with.
How can someone politely insist to communicate in writing only without offending someone and how should they proceed if the person insists on verbal communication instead?
If you want to remedy this with minimal fuss, my suggestions are further below.
But i find the response she is considering sending interesting:
I'm insisting on written communication because I've lost all faith in anything that any of your employees says. I already telephoned with three employees who all alleged different, conflicting reasons for the wrong calculation: a waste of my time. You appear to refuse email because you're trying to shirk leaving a written record that'd be evidence if your employees fib again.
She believes she needs to qualify her request, as though it was unreasonable and she must now elaborate upon it. If she were asking to communicate via carrier pidgeon, yes, by all means, provide a reason why the party should accommodate your request. But when that request is essentially the standard modality of communication, unless that person has a health-related impediment, I would have zero flexibility.
I'm not willing to email. I still think that verbal discussion over the telephone is fastest.
A more professional response would avoid making accusations about lying. Instead, regardless of who is at fault, what you have been trying so far is not working:
Yes, telephone is faster. Unfortunately, we've experienced a number of
mistakes and miscommunications. So instead of focussing on resolving
this quickly, I suggest we try to resolve it correctly. And you may
just find that this is actually the fastest method.
However, there is something about the following that irks me:
I'm not willing to email.
Regardless... perhaps your cousin is more polite and less prideful than me.
I had a similar experience about two weeks ago, for precisely the same reasons. I really wanted a paper trail because I knew it would come back to bite them. I think this is the answer your cousin is actually looking for:
Them: Can we please arrange a me so we can discuss this over the
phone?
Me: If it's not too much trouble I would prefer to correspond via email. However I am in no particular rush so feel free to respond at
your convenience.
Them: I believe it would be easier to discuss this over the phone so
there aren’t any misunderstandings and no further delays. [Accepts request and continues]
... [All subsequent correspondence via e-mail]
...
One alternative method that might be to bend the truth without lying:
I have hearing difficulties, and I believe it has already lead to a number of misunderstandings. If it's not too much trouble, I would prefer to correspond via e-mail.
Imagine now, if the employee said: "well, because I only communicate over the phone, it will be a lot trouble. I tell you what, when you resolve your hearing problems, call me get back and I'll sort this out for you in a FLASH"
LOL. I would love to see the outcome of a performance review where their boss saw that.